| L(s) = 1 | − 12·7-s − 4·13-s − 4·25-s − 24·31-s − 24·37-s − 24·43-s + 58·49-s − 4·61-s − 12·67-s + 24·73-s + 36·79-s + 48·91-s − 28·97-s + 108·103-s + 48·109-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 34·169-s + 173-s + ⋯ |
| L(s) = 1 | − 4.53·7-s − 1.10·13-s − 4/5·25-s − 4.31·31-s − 3.94·37-s − 3.65·43-s + 58/7·49-s − 0.512·61-s − 1.46·67-s + 2.80·73-s + 4.05·79-s + 5.03·91-s − 2.84·97-s + 10.6·103-s + 4.59·109-s + 2.54·121-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 + 2.61·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.742455371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.742455371\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 4 T^{2} + 2 p T^{4} - 176 T^{6} - 989 T^{8} - 176 p^{2} T^{10} + 2 p^{5} T^{12} + 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( ( 1 + 6 T + 25 T^{2} + 78 T^{3} + 204 T^{4} + 78 p T^{5} + 25 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 - 28 T^{2} + 394 T^{4} - 4144 T^{6} + 42595 T^{8} - 4144 p^{2} T^{10} + 394 p^{4} T^{12} - 28 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( ( 1 + 2 T - 11 T^{2} - 22 T^{3} + 4 T^{4} - 22 p T^{5} - 11 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 20 T^{2} + 246 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 + 20 T^{2} - 326 T^{4} - 6640 T^{6} + 35635 T^{8} - 6640 p^{2} T^{10} - 326 p^{4} T^{12} + 20 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( 1 + 52 T^{2} + 1114 T^{4} - 4784 T^{6} - 453245 T^{8} - 4784 p^{2} T^{10} + 1114 p^{4} T^{12} + 52 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3891 T^{4} + 696 p T^{5} + 106 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( 1 + 100 T^{2} + 4906 T^{4} + 173200 T^{6} + 6313075 T^{8} + 173200 p^{2} T^{10} + 4906 p^{4} T^{12} + 100 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 12 T + 130 T^{2} + 984 T^{3} + 6939 T^{4} + 984 p T^{5} + 130 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 76 T^{2} + 1114 T^{4} - 18544 T^{6} + 4095379 T^{8} - 18544 p^{2} T^{10} + 1114 p^{4} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 20 T^{2} - 1194 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 28 T^{2} - 4022 T^{4} + 60368 T^{6} + 8251171 T^{8} + 60368 p^{2} T^{10} - 4022 p^{4} T^{12} - 28 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 2 T - 11 T^{2} - 214 T^{3} - 3740 T^{4} - 214 p T^{5} - 11 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 6 T + 85 T^{2} + 438 T^{3} + 1644 T^{4} + 438 p T^{5} + 85 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 6 T + 107 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 18 T + 193 T^{2} - 1530 T^{3} + 9516 T^{4} - 1530 p T^{5} + 193 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 268 T^{2} + 40858 T^{4} - 4606384 T^{6} + 416639299 T^{8} - 4606384 p^{2} T^{10} + 40858 p^{4} T^{12} - 268 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 308 T^{2} + 39126 T^{4} - 308 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.58415818743394112554873202381, −3.53518287124088793731948816754, −3.46421133629321887354383427454, −3.32586194256858391801452242249, −3.27606275356469846836111920019, −3.26999931223329682549610642879, −3.26773249999508193622474168558, −2.93668811663237612513489289489, −2.78472338743213677801880109541, −2.63791182142274488493094888360, −2.49103777139732255261172446623, −2.24767808248478715270487245374, −2.19706150752174516827524412933, −2.03635068990452131563533694870, −1.92653141635973633019535444798, −1.78950819188395781485461708774, −1.75602049487908321683763271274, −1.58655369161374424983736896236, −1.56711439493954975518367415386, −1.11869130187442382643334747618, −0.59613998304989793285109488810, −0.54417627115254610221453673117, −0.43115137269239682622422149310, −0.37099339749107978956030171642, −0.32209650684622025498821876561,
0.32209650684622025498821876561, 0.37099339749107978956030171642, 0.43115137269239682622422149310, 0.54417627115254610221453673117, 0.59613998304989793285109488810, 1.11869130187442382643334747618, 1.56711439493954975518367415386, 1.58655369161374424983736896236, 1.75602049487908321683763271274, 1.78950819188395781485461708774, 1.92653141635973633019535444798, 2.03635068990452131563533694870, 2.19706150752174516827524412933, 2.24767808248478715270487245374, 2.49103777139732255261172446623, 2.63791182142274488493094888360, 2.78472338743213677801880109541, 2.93668811663237612513489289489, 3.26773249999508193622474168558, 3.26999931223329682549610642879, 3.27606275356469846836111920019, 3.32586194256858391801452242249, 3.46421133629321887354383427454, 3.53518287124088793731948816754, 3.58415818743394112554873202381
Plot not available for L-functions of degree greater than 10.
