| L(s) = 1 | − 2.13·2-s + 2.54·4-s − 3.01·7-s − 1.17·8-s − 5.51·11-s − 0.501·13-s + 6.43·14-s − 2.59·16-s + 6.61·17-s + 4.42·19-s + 11.7·22-s + 1.67·23-s + 1.06·26-s − 7.69·28-s + 1.51·29-s − 9.00·31-s + 7.88·32-s − 14.1·34-s − 37-s − 9.43·38-s − 3.32·41-s + 3.05·43-s − 14.0·44-s − 3.56·46-s + 3.80·47-s + 2.11·49-s − 1.27·52-s + ⋯ |
| L(s) = 1 | − 1.50·2-s + 1.27·4-s − 1.14·7-s − 0.414·8-s − 1.66·11-s − 0.139·13-s + 1.72·14-s − 0.649·16-s + 1.60·17-s + 1.01·19-s + 2.50·22-s + 0.348·23-s + 0.209·26-s − 1.45·28-s + 0.281·29-s − 1.61·31-s + 1.39·32-s − 2.42·34-s − 0.164·37-s − 1.53·38-s − 0.519·41-s + 0.466·43-s − 2.11·44-s − 0.526·46-s + 0.554·47-s + 0.302·49-s − 0.177·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 2.13T + 2T^{2} \) |
| 7 | \( 1 + 3.01T + 7T^{2} \) |
| 11 | \( 1 + 5.51T + 11T^{2} \) |
| 13 | \( 1 + 0.501T + 13T^{2} \) |
| 17 | \( 1 - 6.61T + 17T^{2} \) |
| 19 | \( 1 - 4.42T + 19T^{2} \) |
| 23 | \( 1 - 1.67T + 23T^{2} \) |
| 29 | \( 1 - 1.51T + 29T^{2} \) |
| 31 | \( 1 + 9.00T + 31T^{2} \) |
| 41 | \( 1 + 3.32T + 41T^{2} \) |
| 43 | \( 1 - 3.05T + 43T^{2} \) |
| 47 | \( 1 - 3.80T + 47T^{2} \) |
| 53 | \( 1 + 6.03T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 2.46T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 8.36T + 83T^{2} \) |
| 89 | \( 1 - 7.41T + 89T^{2} \) |
| 97 | \( 1 - 5.75T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.42305130548665229785709521180, −7.31679231788128755045864214512, −6.22344536819059283702552986877, −5.51478627415154036038768284070, −4.85557479968172960942980616688, −3.45066018863158774627472543455, −2.99350810558832778163430922106, −2.02606156674922015501074635828, −0.903468527148951983452151192247, 0,
0.903468527148951983452151192247, 2.02606156674922015501074635828, 2.99350810558832778163430922106, 3.45066018863158774627472543455, 4.85557479968172960942980616688, 5.51478627415154036038768284070, 6.22344536819059283702552986877, 7.31679231788128755045864214512, 7.42305130548665229785709521180
