| L(s) = 1 | + 2.58·2-s + 4.69·4-s − 1.44·5-s − 1.96·7-s + 6.96·8-s − 3.73·10-s − 2.85·11-s − 3.84·13-s − 5.08·14-s + 8.62·16-s − 1.30·17-s + 3.53·19-s − 6.76·20-s − 7.38·22-s − 4.20·23-s − 2.91·25-s − 9.94·26-s − 9.21·28-s + 1.10·29-s − 3.53·31-s + 8.39·32-s − 3.37·34-s + 2.83·35-s − 6.61·37-s + 9.13·38-s − 10.0·40-s − 0.671·41-s + ⋯ |
| L(s) = 1 | + 1.82·2-s + 2.34·4-s − 0.645·5-s − 0.742·7-s + 2.46·8-s − 1.18·10-s − 0.861·11-s − 1.06·13-s − 1.35·14-s + 2.15·16-s − 0.316·17-s + 0.810·19-s − 1.51·20-s − 1.57·22-s − 0.877·23-s − 0.583·25-s − 1.95·26-s − 1.74·28-s + 0.205·29-s − 0.634·31-s + 1.48·32-s − 0.578·34-s + 0.479·35-s − 1.08·37-s + 1.48·38-s − 1.58·40-s − 0.104·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4527 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 2 | \( 1 - 2.58T + 2T^{2} \) |
| 5 | \( 1 + 1.44T + 5T^{2} \) |
| 7 | \( 1 + 1.96T + 7T^{2} \) |
| 11 | \( 1 + 2.85T + 11T^{2} \) |
| 13 | \( 1 + 3.84T + 13T^{2} \) |
| 17 | \( 1 + 1.30T + 17T^{2} \) |
| 19 | \( 1 - 3.53T + 19T^{2} \) |
| 23 | \( 1 + 4.20T + 23T^{2} \) |
| 29 | \( 1 - 1.10T + 29T^{2} \) |
| 31 | \( 1 + 3.53T + 31T^{2} \) |
| 37 | \( 1 + 6.61T + 37T^{2} \) |
| 41 | \( 1 + 0.671T + 41T^{2} \) |
| 43 | \( 1 + 4.32T + 43T^{2} \) |
| 47 | \( 1 - 0.378T + 47T^{2} \) |
| 53 | \( 1 - 7.13T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 6.51T + 61T^{2} \) |
| 67 | \( 1 + 7.55T + 67T^{2} \) |
| 71 | \( 1 + 0.0744T + 71T^{2} \) |
| 73 | \( 1 - 3.29T + 73T^{2} \) |
| 79 | \( 1 - 8.57T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 - 17.5T + 89T^{2} \) |
| 97 | \( 1 + 8.01T + 97T^{2} \) |
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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
−7.53442225736267395616577202960, −7.14367393401188940549614534362, −6.35038808538235034196279151094, −5.50453238700594309388474363353, −5.03930596979110469753224200700, −4.15587391692904573049260167806, −3.50599113313957196341148950422, −2.79508156283973917907269267510, −1.98382529019382990594861369904, 0,
1.98382529019382990594861369904, 2.79508156283973917907269267510, 3.50599113313957196341148950422, 4.15587391692904573049260167806, 5.03930596979110469753224200700, 5.50453238700594309388474363353, 6.35038808538235034196279151094, 7.14367393401188940549614534362, 7.53442225736267395616577202960
