| L(s) = 1 | − 2.09·2-s + 2.39·4-s − 3.55·5-s − 3.93·7-s − 0.836·8-s + 7.45·10-s − 4.67·11-s − 0.747·13-s + 8.25·14-s − 3.04·16-s + 2.98·17-s − 3.50·19-s − 8.52·20-s + 9.79·22-s + 23-s + 7.62·25-s + 1.56·26-s − 9.44·28-s − 6.56·29-s − 5.82·31-s + 8.05·32-s − 6.25·34-s + 13.9·35-s + 7.36·37-s + 7.35·38-s + 2.97·40-s − 41-s + ⋯ |
| L(s) = 1 | − 1.48·2-s + 1.19·4-s − 1.58·5-s − 1.48·7-s − 0.295·8-s + 2.35·10-s − 1.40·11-s − 0.207·13-s + 2.20·14-s − 0.760·16-s + 0.723·17-s − 0.804·19-s − 1.90·20-s + 2.08·22-s + 0.208·23-s + 1.52·25-s + 0.307·26-s − 1.78·28-s − 1.21·29-s − 1.04·31-s + 1.42·32-s − 1.07·34-s + 2.36·35-s + 1.21·37-s + 1.19·38-s + 0.469·40-s − 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8487 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8487 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 2.09T + 2T^{2} \) |
| 5 | \( 1 + 3.55T + 5T^{2} \) |
| 7 | \( 1 + 3.93T + 7T^{2} \) |
| 11 | \( 1 + 4.67T + 11T^{2} \) |
| 13 | \( 1 + 0.747T + 13T^{2} \) |
| 17 | \( 1 - 2.98T + 17T^{2} \) |
| 19 | \( 1 + 3.50T + 19T^{2} \) |
| 29 | \( 1 + 6.56T + 29T^{2} \) |
| 31 | \( 1 + 5.82T + 31T^{2} \) |
| 37 | \( 1 - 7.36T + 37T^{2} \) |
| 43 | \( 1 + 3.51T + 43T^{2} \) |
| 47 | \( 1 + 6.59T + 47T^{2} \) |
| 53 | \( 1 + 0.0352T + 53T^{2} \) |
| 59 | \( 1 - 9.93T + 59T^{2} \) |
| 61 | \( 1 + 7.37T + 61T^{2} \) |
| 67 | \( 1 - 8.02T + 67T^{2} \) |
| 71 | \( 1 - 0.297T + 71T^{2} \) |
| 73 | \( 1 + 9.59T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 5.90T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−7.64732042884035933408906321631, −7.13459403473778385012169171880, −6.43103525220258724552477437024, −5.47084367638098525126744663052, −4.52519128493042823751288750739, −3.63204454022283741185331033255, −3.04590905920369475209812356100, −2.09815285474330481602786742451, −0.59741487739784588976912351520, 0,
0.59741487739784588976912351520, 2.09815285474330481602786742451, 3.04590905920369475209812356100, 3.63204454022283741185331033255, 4.52519128493042823751288750739, 5.47084367638098525126744663052, 6.43103525220258724552477437024, 7.13459403473778385012169171880, 7.64732042884035933408906321631
