| L(s) = 1 | − 2-s + 8·3-s + 8·4-s + 5·5-s − 8·6-s − 23·8-s + 27·9-s − 5·10-s − 12·11-s + 64·12-s − 156·13-s + 40·15-s + 23·16-s + 94·17-s − 27·18-s − 40·19-s + 40·20-s + 12·22-s − 32·23-s − 184·24-s + 156·26-s + 136·27-s − 100·29-s − 40·30-s + 248·31-s − 184·32-s − 96·33-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 1.53·3-s + 4-s + 0.447·5-s − 0.544·6-s − 1.01·8-s + 9-s − 0.158·10-s − 0.328·11-s + 1.53·12-s − 3.32·13-s + 0.688·15-s + 0.359·16-s + 1.34·17-s − 0.353·18-s − 0.482·19-s + 0.447·20-s + 0.116·22-s − 0.290·23-s − 1.56·24-s + 1.17·26-s + 0.969·27-s − 0.640·29-s − 0.243·30-s + 1.43·31-s − 1.01·32-s − 0.506·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.600401374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.600401374\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T - 7 T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 8 T + 37 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 12 T - 1187 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 6 p T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 94 T + 3923 T^{2} - 94 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 40 T - 5259 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 32 T - 11143 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 50 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 p T + 33 p^{2} T^{2} - 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 434 T + 137703 T^{2} - 434 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 402 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 68 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 536 T + 183473 T^{2} + 536 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 22 T - 148393 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 560 T + 108221 T^{2} - 560 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 278 T - 149697 T^{2} - 278 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 164 T - 273867 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 672 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 82 T - 382293 T^{2} + 82 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 1000 T + 506961 T^{2} - 1000 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 448 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 870 T + 51931 T^{2} - 870 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1026 T + p^{3} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.22275038645610997421244524900, −11.40768973904302287485065498208, −10.91243832959521934611217742355, −9.971577241664115524724598188119, −9.968837402293357942852366316704, −9.499636349964238753460230244027, −9.205968650696464005214195720912, −8.274653685467574486814592392118, −8.003675894353149781834667663402, −7.51133658554728079375378808280, −7.18692000107711907117495923723, −6.44801503513919454298746769715, −5.92618307130891297558900212857, −5.07310103018659456139174632648, −4.61724959930536068611984472531, −3.55566665809161712236236601918, −2.74305686494196276188366295539, −2.43558012446739955966055484620, −2.22014920806682429216604940945, −0.68733579840580643744317448991,
0.68733579840580643744317448991, 2.22014920806682429216604940945, 2.43558012446739955966055484620, 2.74305686494196276188366295539, 3.55566665809161712236236601918, 4.61724959930536068611984472531, 5.07310103018659456139174632648, 5.92618307130891297558900212857, 6.44801503513919454298746769715, 7.18692000107711907117495923723, 7.51133658554728079375378808280, 8.003675894353149781834667663402, 8.274653685467574486814592392118, 9.205968650696464005214195720912, 9.499636349964238753460230244027, 9.968837402293357942852366316704, 9.971577241664115524724598188119, 10.91243832959521934611217742355, 11.40768973904302287485065498208, 12.22275038645610997421244524900
