| L(s) = 1 | − 4·2-s + 6.88·3-s + 16·4-s + 25·5-s − 27.5·6-s − 45.2·7-s − 64·8-s − 195.·9-s − 100·10-s + 460.·11-s + 110.·12-s + 403.·13-s + 181.·14-s + 172.·15-s + 256·16-s − 2.06e3·17-s + 782.·18-s − 2.21e3·19-s + 400·20-s − 311.·21-s − 1.84e3·22-s + 529·23-s − 440.·24-s + 625·25-s − 1.61e3·26-s − 3.01e3·27-s − 724.·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.441·3-s + 0.5·4-s + 0.447·5-s − 0.312·6-s − 0.349·7-s − 0.353·8-s − 0.805·9-s − 0.316·10-s + 1.14·11-s + 0.220·12-s + 0.662·13-s + 0.246·14-s + 0.197·15-s + 0.250·16-s − 1.73·17-s + 0.569·18-s − 1.40·19-s + 0.223·20-s − 0.154·21-s − 0.811·22-s + 0.208·23-s − 0.156·24-s + 0.200·25-s − 0.468·26-s − 0.797·27-s − 0.174·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 5 | \( 1 - 25T \) |
| 23 | \( 1 - 529T \) |
| good | 3 | \( 1 - 6.88T + 243T^{2} \) |
| 7 | \( 1 + 45.2T + 1.68e4T^{2} \) |
| 11 | \( 1 - 460.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 403.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.06e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.21e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 375.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.97e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.09e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.68e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.67e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.38e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.69e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.48e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.70e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.00e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.93e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.62e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.25e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.11e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.22e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.21e5T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.89363067258704405854525246153, −9.593074349359755346992978957359, −8.902315202202561352544097061814, −8.247630648277694049544144531411, −6.63927606001474347996440703152, −6.15753435195301283533337661753, −4.29151535494507437137882419567, −2.86323014369633485745373358534, −1.67709566603657981380043793457, 0,
1.67709566603657981380043793457, 2.86323014369633485745373358534, 4.29151535494507437137882419567, 6.15753435195301283533337661753, 6.63927606001474347996440703152, 8.247630648277694049544144531411, 8.902315202202561352544097061814, 9.593074349359755346992978957359, 10.89363067258704405854525246153
