| L(s) = 1 | − 3-s + 9-s − 2·11-s − 13-s + 3·17-s − 23-s − 27-s − 5·29-s + 7·31-s + 2·33-s + 2·37-s + 39-s − 7·41-s − 11·43-s − 8·47-s − 3·51-s + 53-s − 5·59-s + 3·61-s − 12·67-s + 69-s − 12·71-s − 6·73-s − 10·79-s + 81-s + 11·83-s + 5·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 0.727·17-s − 0.208·23-s − 0.192·27-s − 0.928·29-s + 1.25·31-s + 0.348·33-s + 0.328·37-s + 0.160·39-s − 1.09·41-s − 1.67·43-s − 1.16·47-s − 0.420·51-s + 0.137·53-s − 0.650·59-s + 0.384·61-s − 1.46·67-s + 0.120·69-s − 1.42·71-s − 0.702·73-s − 1.12·79-s + 1/9·81-s + 1.20·83-s + 0.536·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8874687858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8874687858\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−14.42624968651241, −13.67961754081023, −13.23245951810758, −12.95398389261853, −12.11238116612490, −11.82221153872033, −11.46113212671618, −10.64273378802419, −10.26259978392624, −9.894625963728385, −9.281661099314938, −8.581977844599842, −8.008036356938591, −7.606271849888018, −6.940818012107432, −6.407949581673077, −5.823205516347204, −5.272699055129798, −4.772874754009128, −4.209002325544695, −3.310798903936702, −2.930735015600565, −1.936313612052946, −1.369536720968158, −0.3373018454497730,
0.3373018454497730, 1.369536720968158, 1.936313612052946, 2.930735015600565, 3.310798903936702, 4.209002325544695, 4.772874754009128, 5.272699055129798, 5.823205516347204, 6.407949581673077, 6.940818012107432, 7.606271849888018, 8.008036356938591, 8.581977844599842, 9.281661099314938, 9.894625963728385, 10.26259978392624, 10.64273378802419, 11.46113212671618, 11.82221153872033, 12.11238116612490, 12.95398389261853, 13.23245951810758, 13.67961754081023, 14.42624968651241
