| L(s) = 1 | − 1.88·2-s + 1.56·4-s + 1.09·5-s − 4.47·7-s + 0.813·8-s − 2.06·10-s − 4.80·11-s − 2.80·13-s + 8.46·14-s − 4.67·16-s − 5.98·17-s − 3.88·19-s + 1.71·20-s + 9.07·22-s + 0.278·23-s − 3.80·25-s + 5.29·26-s − 7.02·28-s + 1.24·29-s + 6.28·31-s + 7.20·32-s + 11.2·34-s − 4.88·35-s + 11.8·37-s + 7.34·38-s + 0.888·40-s − 1.98·41-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 0.784·4-s + 0.488·5-s − 1.69·7-s + 0.287·8-s − 0.652·10-s − 1.44·11-s − 0.776·13-s + 2.26·14-s − 1.16·16-s − 1.45·17-s − 0.891·19-s + 0.383·20-s + 1.93·22-s + 0.0580·23-s − 0.761·25-s + 1.03·26-s − 1.32·28-s + 0.231·29-s + 1.12·31-s + 1.27·32-s + 1.93·34-s − 0.826·35-s + 1.94·37-s + 1.19·38-s + 0.140·40-s − 0.309·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2601086405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2601086405\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 + 1.88T + 2T^{2} \) |
| 5 | \( 1 - 1.09T + 5T^{2} \) |
| 7 | \( 1 + 4.47T + 7T^{2} \) |
| 11 | \( 1 + 4.80T + 11T^{2} \) |
| 13 | \( 1 + 2.80T + 13T^{2} \) |
| 17 | \( 1 + 5.98T + 17T^{2} \) |
| 19 | \( 1 + 3.88T + 19T^{2} \) |
| 23 | \( 1 - 0.278T + 23T^{2} \) |
| 29 | \( 1 - 1.24T + 29T^{2} \) |
| 31 | \( 1 - 6.28T + 31T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 + 1.98T + 41T^{2} \) |
| 43 | \( 1 + 9.05T + 43T^{2} \) |
| 47 | \( 1 + 6.27T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 5.47T + 59T^{2} \) |
| 61 | \( 1 - 3.52T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 5.61T + 71T^{2} \) |
| 73 | \( 1 + 2.99T + 73T^{2} \) |
| 79 | \( 1 - 4.90T + 79T^{2} \) |
| 83 | \( 1 - 4.56T + 83T^{2} \) |
| 89 | \( 1 - 1.20T + 89T^{2} \) |
| 97 | \( 1 + 5.55T + 97T^{2} \) |
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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
−9.240160002263845876805179518486, −8.389691728638086269207911423873, −7.75429614452553815825179744667, −6.71137986283931321204114063468, −6.39524487472137466526026130684, −5.17518450569400094400207160560, −4.19168600570077798165703402276, −2.72865171507555024908451878006, −2.21561703534991532961875718116, −0.37755825352898693789115362840,
0.37755825352898693789115362840, 2.21561703534991532961875718116, 2.72865171507555024908451878006, 4.19168600570077798165703402276, 5.17518450569400094400207160560, 6.39524487472137466526026130684, 6.71137986283931321204114063468, 7.75429614452553815825179744667, 8.389691728638086269207911423873, 9.240160002263845876805179518486
