| L(s) = 1 | + 4·4-s + 4·16-s + 16·19-s − 16·41-s − 64·59-s + 24·61-s − 16·64-s + 64·76-s + 32·79-s − 2·81-s + 16·101-s + 72·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 64·164-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2·4-s + 16-s + 3.67·19-s − 2.49·41-s − 8.33·59-s + 3.07·61-s − 2·64-s + 7.34·76-s + 3.60·79-s − 2/9·81-s + 1.59·101-s + 6.54·121-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 − 4.99·164-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.432290639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.432290639\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | \( ( 1 + T^{4} )^{2} \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2 p^{2} T^{4} + 7011 T^{8} - 2 p^{6} T^{12} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( 1 - 194 T^{4} + 171 p^{2} T^{8} - 194 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | \( ( 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 + 68 T^{4} - 434490 T^{8} + 68 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 60 T^{2} + 1814 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 110 T^{2} + 4899 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 + 868 T^{4} + 2707878 T^{8} + 868 p^{4} T^{12} + p^{8} T^{16} \) |
| 41 | \( ( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 + 3358 T^{4} + 8633475 T^{8} + 3358 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( 1 + 5188 T^{4} + 13723398 T^{8} + 5188 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( 1 + 5348 T^{4} + 16710438 T^{8} + 5348 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 16 T + 170 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 3 T + p T^{2} )^{8} \) |
| 67 | \( 1 - 6818 T^{4} + 25780611 T^{8} - 6818 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 188 T^{2} + 17190 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 6242 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( 1 - 284 T^{4} - 90968346 T^{8} - 284 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 - 3842 T^{4} + 131200515 T^{8} - 3842 p^{4} T^{12} + p^{8} T^{16} \) |
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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
−5.39966334786721538441851554985, −4.97637969915031303869463374087, −4.93455500706823304270614398985, −4.71745284857525853345750857197, −4.70433408026730913379935274600, −4.66002522645508902310423901210, −4.63842603732416048319774398786, −4.15779414298777680929023556903, −4.06911851531989675583786129916, −3.80052576868758004327729803635, −3.41108022194449428665309711009, −3.29619556753170535485728675153, −3.22439725721287402853970558972, −3.17416933566378841502331827245, −3.15591942293845268274111660768, −3.12776389750713360144645410640, −2.58457900479375986576438811531, −2.29727584641185194499338909497, −2.25269936065356816363764482463, −1.83813186662703678092373766035, −1.78048616440873034354294552684, −1.77039408276890328389779016259, −1.30192589175606002282227612411, −0.953076200547883476621943553750, −0.56835596978907116129257804668,
0.56835596978907116129257804668, 0.953076200547883476621943553750, 1.30192589175606002282227612411, 1.77039408276890328389779016259, 1.78048616440873034354294552684, 1.83813186662703678092373766035, 2.25269936065356816363764482463, 2.29727584641185194499338909497, 2.58457900479375986576438811531, 3.12776389750713360144645410640, 3.15591942293845268274111660768, 3.17416933566378841502331827245, 3.22439725721287402853970558972, 3.29619556753170535485728675153, 3.41108022194449428665309711009, 3.80052576868758004327729803635, 4.06911851531989675583786129916, 4.15779414298777680929023556903, 4.63842603732416048319774398786, 4.66002522645508902310423901210, 4.70433408026730913379935274600, 4.71745284857525853345750857197, 4.93455500706823304270614398985, 4.97637969915031303869463374087, 5.39966334786721538441851554985
Plot not available for L-functions of degree greater than 10.
