| L(s) = 1 | − 1.93·2-s − 2.41·3-s + 1.74·4-s + 4.67·6-s − 1.46·7-s + 0.487·8-s + 2.82·9-s − 2.89·11-s − 4.22·12-s + 6.12·13-s + 2.82·14-s − 4.44·16-s + 6.29·17-s − 5.47·18-s + 3.52·21-s + 5.60·22-s − 0.508·23-s − 1.17·24-s − 11.8·26-s + 0.414·27-s − 2.55·28-s + 1.72·29-s − 8.44·31-s + 7.62·32-s + 6.98·33-s − 12.1·34-s + 4.94·36-s + ⋯ |
| L(s) = 1 | − 1.36·2-s − 1.39·3-s + 0.874·4-s + 1.90·6-s − 0.552·7-s + 0.172·8-s + 0.942·9-s − 0.872·11-s − 1.21·12-s + 1.69·13-s + 0.755·14-s − 1.11·16-s + 1.52·17-s − 1.29·18-s + 0.769·21-s + 1.19·22-s − 0.106·23-s − 0.240·24-s − 2.32·26-s + 0.0796·27-s − 0.482·28-s + 0.319·29-s − 1.51·31-s + 1.34·32-s + 1.21·33-s − 2.09·34-s + 0.824·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.93T + 2T^{2} \) |
| 3 | \( 1 + 2.41T + 3T^{2} \) |
| 7 | \( 1 + 1.46T + 7T^{2} \) |
| 11 | \( 1 + 2.89T + 11T^{2} \) |
| 13 | \( 1 - 6.12T + 13T^{2} \) |
| 17 | \( 1 - 6.29T + 17T^{2} \) |
| 23 | \( 1 + 0.508T + 23T^{2} \) |
| 29 | \( 1 - 1.72T + 29T^{2} \) |
| 31 | \( 1 + 8.44T + 31T^{2} \) |
| 37 | \( 1 - 3.13T + 37T^{2} \) |
| 41 | \( 1 - 7.44T + 41T^{2} \) |
| 43 | \( 1 + 6.90T + 43T^{2} \) |
| 47 | \( 1 - 0.316T + 47T^{2} \) |
| 53 | \( 1 - 4.34T + 53T^{2} \) |
| 59 | \( 1 - 2.25T + 59T^{2} \) |
| 61 | \( 1 + 6.29T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 6.63T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 4.94T + 83T^{2} \) |
| 89 | \( 1 + 5.35T + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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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
−7.54861226183501367838840483916, −6.74300924597096588127365477266, −6.03905416594045330968771041262, −5.63126797612682117757505159282, −4.81698042557948649596751732172, −3.82948176049907503938634448608, −2.94780927350815169185835501018, −1.60274279906773632899235459262, −0.899854594658749537795881587947, 0,
0.899854594658749537795881587947, 1.60274279906773632899235459262, 2.94780927350815169185835501018, 3.82948176049907503938634448608, 4.81698042557948649596751732172, 5.63126797612682117757505159282, 6.03905416594045330968771041262, 6.74300924597096588127365477266, 7.54861226183501367838840483916
