| L(s) = 1 | + 2·9-s − 2·11-s − 14·19-s − 8·29-s + 38·31-s + 50·41-s + 10·49-s + 2·59-s − 10·61-s + 2·71-s + 4·79-s + 11·81-s − 12·89-s − 4·99-s − 22·101-s + 26·109-s − 37·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 0.603·11-s − 3.21·19-s − 1.48·29-s + 6.82·31-s + 7.80·41-s + 10/7·49-s + 0.260·59-s − 1.28·61-s + 0.237·71-s + 0.450·79-s + 11/9·81-s − 1.27·89-s − 0.402·99-s − 2.18·101-s + 2.49·109-s − 3.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 5^{20} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 5^{20} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(34.70815739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(34.70815739\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( ( 1 + T^{2} )^{5} \) |
| good | 3 | \( 1 - 2 T^{2} - 7 T^{4} + 11 p T^{6} + 22 p T^{8} - 518 T^{10} + 22 p^{3} T^{12} + 11 p^{5} T^{14} - 7 p^{6} T^{16} - 2 p^{8} T^{18} + p^{10} T^{20} \) |
| 7 | \( 1 - 10 T^{2} + 113 T^{4} - 655 T^{6} + 8930 T^{8} - 58406 T^{10} + 8930 p^{2} T^{12} - 655 p^{4} T^{14} + 113 p^{6} T^{16} - 10 p^{8} T^{18} + p^{10} T^{20} \) |
| 11 | \( ( 1 + T + 20 T^{2} + 16 T^{3} + 227 T^{4} + 174 T^{5} + 227 p T^{6} + 16 p^{2} T^{7} + 20 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 13 | \( 1 - 22 T^{2} + 289 T^{4} - 1207 T^{6} - 32486 T^{8} + 746834 T^{10} - 32486 p^{2} T^{12} - 1207 p^{4} T^{14} + 289 p^{6} T^{16} - 22 p^{8} T^{18} + p^{10} T^{20} \) |
| 17 | \( 1 - 62 T^{2} + 1413 T^{4} - 19719 T^{6} + 460346 T^{8} - 10620406 T^{10} + 460346 p^{2} T^{12} - 19719 p^{4} T^{14} + 1413 p^{6} T^{16} - 62 p^{8} T^{18} + p^{10} T^{20} \) |
| 19 | \( ( 1 + 7 T + 54 T^{2} + 352 T^{3} + 1949 T^{4} + 7810 T^{5} + 1949 p T^{6} + 352 p^{2} T^{7} + 54 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 29 | \( ( 1 + 4 T + 104 T^{2} + 500 T^{3} + 4907 T^{4} + 22264 T^{5} + 4907 p T^{6} + 500 p^{2} T^{7} + 104 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 31 | \( ( 1 - 19 T + 227 T^{2} - 2173 T^{3} + 16253 T^{4} - 98336 T^{5} + 16253 p T^{6} - 2173 p^{2} T^{7} + 227 p^{3} T^{8} - 19 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 37 | \( 1 - 273 T^{2} + 961 p T^{4} - 2140 p^{2} T^{6} + 169925746 T^{8} - 7270625478 T^{10} + 169925746 p^{2} T^{12} - 2140 p^{6} T^{14} + 961 p^{7} T^{16} - 273 p^{8} T^{18} + p^{10} T^{20} \) |
| 41 | \( ( 1 - 25 T + 417 T^{2} - 4753 T^{3} + 42913 T^{4} - 303514 T^{5} + 42913 p T^{6} - 4753 p^{2} T^{7} + 417 p^{3} T^{8} - 25 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 43 | \( ( 1 - p T^{2} )^{10} \) |
| 47 | \( 1 - 129 T^{2} + 6241 T^{4} - 171172 T^{6} + 2184446 T^{8} + 55511114 T^{10} + 2184446 p^{2} T^{12} - 171172 p^{4} T^{14} + 6241 p^{6} T^{16} - 129 p^{8} T^{18} + p^{10} T^{20} \) |
| 53 | \( 1 - 201 T^{2} + 505 p T^{4} - 2489148 T^{6} + 184152042 T^{8} - 10758932438 T^{10} + 184152042 p^{2} T^{12} - 2489148 p^{4} T^{14} + 505 p^{7} T^{16} - 201 p^{8} T^{18} + p^{10} T^{20} \) |
| 59 | \( ( 1 - T + 153 T^{2} + 14236 T^{4} - 6606 T^{5} + 14236 p T^{6} + 153 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 61 | \( ( 1 + 5 T + 190 T^{2} + 652 T^{3} + 18257 T^{4} + 49998 T^{5} + 18257 p T^{6} + 652 p^{2} T^{7} + 190 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 67 | \( 1 - 337 T^{2} + 56833 T^{4} - 6372324 T^{6} + 545879206 T^{8} - 39174901686 T^{10} + 545879206 p^{2} T^{12} - 6372324 p^{4} T^{14} + 56833 p^{6} T^{16} - 337 p^{8} T^{18} + p^{10} T^{20} \) |
| 71 | \( ( 1 - T + 141 T^{2} + 123 T^{3} + 12967 T^{4} + 23580 T^{5} + 12967 p T^{6} + 123 p^{2} T^{7} + 141 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 73 | \( 1 - 413 T^{2} + 84329 T^{4} - 11643332 T^{6} + 1219588838 T^{8} - 100263599614 T^{10} + 1219588838 p^{2} T^{12} - 11643332 p^{4} T^{14} + 84329 p^{6} T^{16} - 413 p^{8} T^{18} + p^{10} T^{20} \) |
| 79 | \( ( 1 - 2 T + 267 T^{2} - 760 T^{3} + 34506 T^{4} - 94092 T^{5} + 34506 p T^{6} - 760 p^{2} T^{7} + 267 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 83 | \( 1 - 373 T^{2} + 81509 T^{4} - 12399052 T^{6} + 1453736194 T^{8} - 134473058078 T^{10} + 1453736194 p^{2} T^{12} - 12399052 p^{4} T^{14} + 81509 p^{6} T^{16} - 373 p^{8} T^{18} + p^{10} T^{20} \) |
| 89 | \( ( 1 + 6 T + 309 T^{2} + 1624 T^{3} + 45458 T^{4} + 202212 T^{5} + 45458 p T^{6} + 1624 p^{2} T^{7} + 309 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 97 | \( 1 - 267 T^{2} + 53782 T^{4} - 8189842 T^{6} + 1008134697 T^{8} - 107070740038 T^{10} + 1008134697 p^{2} T^{12} - 8189842 p^{4} T^{14} + 53782 p^{6} T^{16} - 267 p^{8} T^{18} + p^{10} T^{20} \) |
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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
−2.78200917625476261562665684948, −2.77983141022125751479333583831, −2.76958646165625515992098057370, −2.68785470002435196219351418629, −2.56699157076202599025529859175, −2.56131912838282152944323167143, −2.28682162308717700497834461142, −2.17451595745106132669904686681, −2.06285935240457617939441509754, −1.89470826237627409253672751881, −1.86562988398739534014016613601, −1.81678688265769728635427076265, −1.80869219076614632485252519508, −1.66935116697542982888828186182, −1.48704176312984594711464798009, −1.39094778762647276571077869526, −1.06603649084009464945521913997, −1.01005192649590272653572104485, −0.892607637749322061266559706866, −0.799596821896219276811284577075, −0.68577344853494979427635592576, −0.63553877813517176601706157036, −0.55996922729282547546059407866, −0.41896685777278383005656231709, −0.20775168096376830664996924429,
0.20775168096376830664996924429, 0.41896685777278383005656231709, 0.55996922729282547546059407866, 0.63553877813517176601706157036, 0.68577344853494979427635592576, 0.799596821896219276811284577075, 0.892607637749322061266559706866, 1.01005192649590272653572104485, 1.06603649084009464945521913997, 1.39094778762647276571077869526, 1.48704176312984594711464798009, 1.66935116697542982888828186182, 1.80869219076614632485252519508, 1.81678688265769728635427076265, 1.86562988398739534014016613601, 1.89470826237627409253672751881, 2.06285935240457617939441509754, 2.17451595745106132669904686681, 2.28682162308717700497834461142, 2.56131912838282152944323167143, 2.56699157076202599025529859175, 2.68785470002435196219351418629, 2.76958646165625515992098057370, 2.77983141022125751479333583831, 2.78200917625476261562665684948
Plot not available for L-functions of degree greater than 10.
