L(s) = 1 | − 5·2-s − 3·3-s + 10·4-s − 7·5-s + 15·6-s − 3·7-s − 5·8-s + 8·9-s + 35·10-s − 30·12-s + 20·13-s + 15·14-s + 21·15-s − 20·16-s − 8·17-s − 40·18-s − 2·19-s − 70·20-s + 9·21-s − 23-s + 15·24-s + 31·25-s − 100·26-s − 11·27-s − 30·28-s − 10·29-s − 105·30-s + ⋯ |
L(s) = 1 | − 3.53·2-s − 1.73·3-s + 5·4-s − 3.13·5-s + 6.12·6-s − 1.13·7-s − 1.76·8-s + 8/3·9-s + 11.0·10-s − 8.66·12-s + 5.54·13-s + 4.00·14-s + 5.42·15-s − 5·16-s − 1.94·17-s − 9.42·18-s − 0.458·19-s − 15.6·20-s + 1.96·21-s − 0.208·23-s + 3.06·24-s + 31/5·25-s − 19.6·26-s − 2.11·27-s − 5.66·28-s − 1.85·29-s − 19.1·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 7^{10} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 7^{10} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.07752770456\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07752770456\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( ( 1 + T + T^{2} )^{5} \) |
| 7 | \( 1 + 3 T + 10 T^{2} + 20 T^{3} - 2 T^{4} + 29 T^{5} - 2 p T^{6} + 20 p^{2} T^{7} + 10 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \) |
| 29 | \( ( 1 + T )^{10} \) |
good | 3 | \( 1 + p T + T^{2} - 10 T^{3} - 20 T^{4} - 7 T^{5} + 58 T^{6} + 131 T^{7} + 64 T^{8} - 55 p T^{9} - 359 T^{10} - 55 p^{2} T^{11} + 64 p^{2} T^{12} + 131 p^{3} T^{13} + 58 p^{4} T^{14} - 7 p^{5} T^{15} - 20 p^{6} T^{16} - 10 p^{7} T^{17} + p^{8} T^{18} + p^{10} T^{19} + p^{10} T^{20} \) |
| 5 | \( 1 + 7 T + 18 T^{2} + 7 T^{3} - 67 T^{4} - 156 T^{5} + 7 p T^{6} + 803 T^{7} + 754 T^{8} - 5437 T^{9} - 20889 T^{10} - 5437 p T^{11} + 754 p^{2} T^{12} + 803 p^{3} T^{13} + 7 p^{5} T^{14} - 156 p^{5} T^{15} - 67 p^{6} T^{16} + 7 p^{7} T^{17} + 18 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) |
| 11 | \( 1 - 2 p T^{2} - 108 T^{3} + 375 T^{4} + 2367 T^{5} + 2916 T^{6} - 39456 T^{7} - 107445 T^{8} + 142767 T^{9} + 2033871 T^{10} + 142767 p T^{11} - 107445 p^{2} T^{12} - 39456 p^{3} T^{13} + 2916 p^{4} T^{14} + 2367 p^{5} T^{15} + 375 p^{6} T^{16} - 108 p^{7} T^{17} - 2 p^{9} T^{18} + p^{10} T^{20} \) |
| 13 | \( ( 1 - 10 T + 93 T^{2} - 513 T^{3} + 2664 T^{4} - 9855 T^{5} + 2664 p T^{6} - 513 p^{2} T^{7} + 93 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 17 | \( 1 + 8 T - 30 T^{2} - 292 T^{3} + 1331 T^{4} + 525 p T^{5} - 32224 T^{6} - 125408 T^{7} + 865243 T^{8} + 1135609 T^{9} - 14927997 T^{10} + 1135609 p T^{11} + 865243 p^{2} T^{12} - 125408 p^{3} T^{13} - 32224 p^{4} T^{14} + 525 p^{6} T^{15} + 1331 p^{6} T^{16} - 292 p^{7} T^{17} - 30 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) |
| 19 | \( 1 + 2 T - 28 T^{2} + 36 T^{3} + 239 T^{4} - 1869 T^{5} + 3316 T^{6} - 1086 T^{7} - 96433 T^{8} + 214923 T^{9} + 596683 T^{10} + 214923 p T^{11} - 96433 p^{2} T^{12} - 1086 p^{3} T^{13} + 3316 p^{4} T^{14} - 1869 p^{5} T^{15} + 239 p^{6} T^{16} + 36 p^{7} T^{17} - 28 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) |
| 23 | \( 1 + T - 75 T^{2} - 302 T^{3} + 3032 T^{4} + 16857 T^{5} - 48532 T^{6} - 514015 T^{7} - 99812 T^{8} + 5176061 T^{9} + 22601625 T^{10} + 5176061 p T^{11} - 99812 p^{2} T^{12} - 514015 p^{3} T^{13} - 48532 p^{4} T^{14} + 16857 p^{5} T^{15} + 3032 p^{6} T^{16} - 302 p^{7} T^{17} - 75 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) |
| 31 | \( 1 + 11 T - 10 T^{2} - 705 T^{3} - 2713 T^{4} + 258 p T^{5} + 99463 T^{6} + 654333 T^{7} + 2865674 T^{8} - 16068147 T^{9} - 218957921 T^{10} - 16068147 p T^{11} + 2865674 p^{2} T^{12} + 654333 p^{3} T^{13} + 99463 p^{4} T^{14} + 258 p^{6} T^{15} - 2713 p^{6} T^{16} - 705 p^{7} T^{17} - 10 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} \) |
| 37 | \( 1 - 8 T - 2 p T^{2} + 420 T^{3} + 4737 T^{4} - 8613 T^{5} - 223524 T^{6} - 116508 T^{7} + 8653143 T^{8} + 7095205 T^{9} - 336141821 T^{10} + 7095205 p T^{11} + 8653143 p^{2} T^{12} - 116508 p^{3} T^{13} - 223524 p^{4} T^{14} - 8613 p^{5} T^{15} + 4737 p^{6} T^{16} + 420 p^{7} T^{17} - 2 p^{9} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) |
| 41 | \( ( 1 - 23 T + 337 T^{2} - 3477 T^{3} + 29189 T^{4} - 200675 T^{5} + 29189 p T^{6} - 3477 p^{2} T^{7} + 337 p^{3} T^{8} - 23 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 43 | \( ( 1 + 3 T + 52 T^{2} + 137 T^{3} + 3085 T^{4} + 17531 T^{5} + 3085 p T^{6} + 137 p^{2} T^{7} + 52 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 47 | \( 1 + 16 T + 12 T^{2} - 944 T^{3} - 1405 T^{4} + 39537 T^{5} + 38546 T^{6} - 941992 T^{7} + 2028553 T^{8} - 11928403 T^{9} - 487573749 T^{10} - 11928403 p T^{11} + 2028553 p^{2} T^{12} - 941992 p^{3} T^{13} + 38546 p^{4} T^{14} + 39537 p^{5} T^{15} - 1405 p^{6} T^{16} - 944 p^{7} T^{17} + 12 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \) |
| 53 | \( 1 + 7 T - 3 p T^{2} - 1610 T^{3} + 13346 T^{4} + 173235 T^{5} - 467908 T^{6} - 10325179 T^{7} - 2598314 T^{8} + 243927971 T^{9} + 1090622757 T^{10} + 243927971 p T^{11} - 2598314 p^{2} T^{12} - 10325179 p^{3} T^{13} - 467908 p^{4} T^{14} + 173235 p^{5} T^{15} + 13346 p^{6} T^{16} - 1610 p^{7} T^{17} - 3 p^{9} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) |
| 59 | \( 1 - 9 T - 175 T^{2} + 1362 T^{3} + 20370 T^{4} - 107499 T^{5} - 1985070 T^{6} + 4843827 T^{7} + 168351006 T^{8} - 111026337 T^{9} - 11159220111 T^{10} - 111026337 p T^{11} + 168351006 p^{2} T^{12} + 4843827 p^{3} T^{13} - 1985070 p^{4} T^{14} - 107499 p^{5} T^{15} + 20370 p^{6} T^{16} + 1362 p^{7} T^{17} - 175 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \) |
| 61 | \( 1 + 15 T - 70 T^{2} - 1619 T^{3} + 6498 T^{4} + 97961 T^{5} - 892312 T^{6} - 4989339 T^{7} + 82929877 T^{8} + 2700054 p T^{9} - 5160608628 T^{10} + 2700054 p^{2} T^{11} + 82929877 p^{2} T^{12} - 4989339 p^{3} T^{13} - 892312 p^{4} T^{14} + 97961 p^{5} T^{15} + 6498 p^{6} T^{16} - 1619 p^{7} T^{17} - 70 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20} \) |
| 67 | \( 1 - 4 T - 145 T^{2} + 90 T^{3} + 10175 T^{4} + 30390 T^{5} - 349805 T^{6} - 40977 p T^{7} + 5519210 T^{8} + 51101871 T^{9} + 268701463 T^{10} + 51101871 p T^{11} + 5519210 p^{2} T^{12} - 40977 p^{4} T^{13} - 349805 p^{4} T^{14} + 30390 p^{5} T^{15} + 10175 p^{6} T^{16} + 90 p^{7} T^{17} - 145 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) |
| 71 | \( ( 1 + 22 T + 406 T^{2} + 5106 T^{3} + 58901 T^{4} + 521437 T^{5} + 58901 p T^{6} + 5106 p^{2} T^{7} + 406 p^{3} T^{8} + 22 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 73 | \( 1 - 118 T^{2} - 452 T^{3} + 6081 T^{4} + 60005 T^{5} + 270680 T^{6} - 4890660 T^{7} - 59485121 T^{8} + 144788235 T^{9} + 5109022011 T^{10} + 144788235 p T^{11} - 59485121 p^{2} T^{12} - 4890660 p^{3} T^{13} + 270680 p^{4} T^{14} + 60005 p^{5} T^{15} + 6081 p^{6} T^{16} - 452 p^{7} T^{17} - 118 p^{8} T^{18} + p^{10} T^{20} \) |
| 79 | \( 1 - 13 T - 193 T^{2} + 2748 T^{3} + 27938 T^{4} - 356307 T^{5} - 2825210 T^{6} + 26899935 T^{7} + 254667098 T^{8} - 944511897 T^{9} - 19793604551 T^{10} - 944511897 p T^{11} + 254667098 p^{2} T^{12} + 26899935 p^{3} T^{13} - 2825210 p^{4} T^{14} - 356307 p^{5} T^{15} + 27938 p^{6} T^{16} + 2748 p^{7} T^{17} - 193 p^{8} T^{18} - 13 p^{9} T^{19} + p^{10} T^{20} \) |
| 83 | \( ( 1 - 28 T + 628 T^{2} - 8973 T^{3} + 112748 T^{4} - 1077883 T^{5} + 112748 p T^{6} - 8973 p^{2} T^{7} + 628 p^{3} T^{8} - 28 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 89 | \( 1 + 17 T + 39 T^{2} - 382 T^{3} - 9760 T^{4} - 233487 T^{5} - 2201065 T^{6} - 8579312 T^{7} + 89338738 T^{8} + 1945154872 T^{9} + 19108773135 T^{10} + 1945154872 p T^{11} + 89338738 p^{2} T^{12} - 8579312 p^{3} T^{13} - 2201065 p^{4} T^{14} - 233487 p^{5} T^{15} - 9760 p^{6} T^{16} - 382 p^{7} T^{17} + 39 p^{8} T^{18} + 17 p^{9} T^{19} + p^{10} T^{20} \) |
| 97 | \( ( 1 - 42 T + 1024 T^{2} - 18106 T^{3} + 247831 T^{4} - 2707201 T^{5} + 247831 p T^{6} - 18106 p^{2} T^{7} + 1024 p^{3} T^{8} - 42 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.14110323070457783537543150769, −4.08132886613802234936508539041, −4.03677650483768899987299587188, −4.03486388851155453730492737484, −3.74158905185707449796251631970, −3.60425973440226220241552597735, −3.53749833900974091040042168261, −3.51202186549112144867401273535, −3.45721348076305865215755778315, −3.38105933494367152194068337580, −3.04848206900724805586044204226, −2.76168929499677774118702486547, −2.65987415526625686169579561304, −2.48049727077483476197382697780, −2.33358799045613421298476240414, −2.00982342812142428317385192032, −1.78696425079981958534911999372, −1.63605964822940966429993954603, −1.53685395416606056083215366510, −1.24732823064923001899545718496, −1.13178229940640331611444409566, −0.851976642188361233399330373561, −0.70051148543055128588751447058, −0.51679547235511003298218947746, −0.29360496097624797365963379682,
0.29360496097624797365963379682, 0.51679547235511003298218947746, 0.70051148543055128588751447058, 0.851976642188361233399330373561, 1.13178229940640331611444409566, 1.24732823064923001899545718496, 1.53685395416606056083215366510, 1.63605964822940966429993954603, 1.78696425079981958534911999372, 2.00982342812142428317385192032, 2.33358799045613421298476240414, 2.48049727077483476197382697780, 2.65987415526625686169579561304, 2.76168929499677774118702486547, 3.04848206900724805586044204226, 3.38105933494367152194068337580, 3.45721348076305865215755778315, 3.51202186549112144867401273535, 3.53749833900974091040042168261, 3.60425973440226220241552597735, 3.74158905185707449796251631970, 4.03486388851155453730492737484, 4.03677650483768899987299587188, 4.08132886613802234936508539041, 4.14110323070457783537543150769
Plot not available for L-functions of degree greater than 10.