| L(s) = 1 | − 1.34·2-s − 0.184·4-s + 5-s − 2.18·7-s + 2.94·8-s − 1.34·10-s + 3.87·11-s + 13-s + 2.94·14-s − 3.59·16-s + 3.71·17-s + 2.41·19-s − 0.184·20-s − 5.22·22-s − 4.75·23-s + 25-s − 1.34·26-s + 0.403·28-s − 3.57·29-s − 10.8·31-s − 1.04·32-s − 5.00·34-s − 2.18·35-s − 7.04·37-s − 3.24·38-s + 2.94·40-s − 1.97·41-s + ⋯ |
| L(s) = 1 | − 0.952·2-s − 0.0923·4-s + 0.447·5-s − 0.825·7-s + 1.04·8-s − 0.426·10-s + 1.16·11-s + 0.277·13-s + 0.786·14-s − 0.899·16-s + 0.901·17-s + 0.553·19-s − 0.0413·20-s − 1.11·22-s − 0.992·23-s + 0.200·25-s − 0.264·26-s + 0.0762·28-s − 0.663·29-s − 1.94·31-s − 0.184·32-s − 0.858·34-s − 0.369·35-s − 1.15·37-s − 0.527·38-s + 0.465·40-s − 0.308·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 1.34T + 2T^{2} \) |
| 7 | \( 1 + 2.18T + 7T^{2} \) |
| 11 | \( 1 - 3.87T + 11T^{2} \) |
| 17 | \( 1 - 3.71T + 17T^{2} \) |
| 19 | \( 1 - 2.41T + 19T^{2} \) |
| 23 | \( 1 + 4.75T + 23T^{2} \) |
| 29 | \( 1 + 3.57T + 29T^{2} \) |
| 31 | \( 1 + 10.8T + 31T^{2} \) |
| 37 | \( 1 + 7.04T + 37T^{2} \) |
| 41 | \( 1 + 1.97T + 41T^{2} \) |
| 43 | \( 1 + 1.14T + 43T^{2} \) |
| 47 | \( 1 + 1.04T + 47T^{2} \) |
| 53 | \( 1 + 1.95T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 - 4.63T + 73T^{2} \) |
| 79 | \( 1 + 17.2T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−7.927291438125169630142662897369, −7.22159876615401146573506209026, −6.56638666884717972023805319112, −5.75649076459094722668343069766, −5.04880642972929575888149508102, −3.82328920181728554818119243979, −3.47664352826851785559778556805, −1.98289511371723640231286974714, −1.25028835046454244985205494872, 0,
1.25028835046454244985205494872, 1.98289511371723640231286974714, 3.47664352826851785559778556805, 3.82328920181728554818119243979, 5.04880642972929575888149508102, 5.75649076459094722668343069766, 6.56638666884717972023805319112, 7.22159876615401146573506209026, 7.927291438125169630142662897369
