| L(s) = 1 | − 3.55·2-s + 4.66·4-s − 7.68·5-s − 2.28·7-s + 11.8·8-s + 27.3·10-s − 8.65·11-s − 56.8·13-s + 8.15·14-s − 79.5·16-s + 67.2·17-s + 115.·19-s − 35.8·20-s + 30.8·22-s − 75.2·23-s − 65.8·25-s + 202.·26-s − 10.6·28-s + 45.2·29-s + 8.72·31-s + 188.·32-s − 239.·34-s + 17.6·35-s − 196.·37-s − 410.·38-s − 91.1·40-s + 428.·41-s + ⋯ |
| L(s) = 1 | − 1.25·2-s + 0.583·4-s − 0.687·5-s − 0.123·7-s + 0.524·8-s + 0.865·10-s − 0.237·11-s − 1.21·13-s + 0.155·14-s − 1.24·16-s + 0.959·17-s + 1.39·19-s − 0.401·20-s + 0.298·22-s − 0.681·23-s − 0.527·25-s + 1.52·26-s − 0.0721·28-s + 0.289·29-s + 0.0505·31-s + 1.04·32-s − 1.20·34-s + 0.0850·35-s − 0.872·37-s − 1.75·38-s − 0.360·40-s + 1.63·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + 43T \) |
| good | 2 | \( 1 + 3.55T + 8T^{2} \) |
| 5 | \( 1 + 7.68T + 125T^{2} \) |
| 7 | \( 1 + 2.28T + 343T^{2} \) |
| 11 | \( 1 + 8.65T + 1.33e3T^{2} \) |
| 13 | \( 1 + 56.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 67.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 115.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 75.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 45.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 8.72T + 2.97e4T^{2} \) |
| 37 | \( 1 + 196.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 428.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 400.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 427.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 473.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 613.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 851.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 501.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 345.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.18e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 589.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.06e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.19e3T + 9.12e5T^{2} \) |
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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
−9.150526207378921153697178934547, −7.955482605026799328280389708965, −7.73326753218343877414260323935, −6.97451885423875332099173532202, −5.62204938840798489892982266806, −4.68129358394392157709891508721, −3.57685218609930869216576829337, −2.35098921072835346353169373726, −1.00697691267201733567002865944, 0,
1.00697691267201733567002865944, 2.35098921072835346353169373726, 3.57685218609930869216576829337, 4.68129358394392157709891508721, 5.62204938840798489892982266806, 6.97451885423875332099173532202, 7.73326753218343877414260323935, 7.955482605026799328280389708965, 9.150526207378921153697178934547
