| L(s) = 1 | − 0.311·2-s + 3-s − 1.90·4-s + 1.90·5-s − 0.311·6-s + 1.21·8-s + 9-s − 0.592·10-s + 2.42·11-s − 1.90·12-s − 2.11·13-s + 1.90·15-s + 3.42·16-s − 5.52·17-s − 0.311·18-s + 7.52·19-s − 3.62·20-s − 0.755·22-s − 5.36·23-s + 1.21·24-s − 1.37·25-s + 0.658·26-s + 27-s − 4.59·29-s − 0.592·30-s − 2.90·31-s − 3.49·32-s + ⋯ |
| L(s) = 1 | − 0.219·2-s + 0.577·3-s − 0.951·4-s + 0.851·5-s − 0.127·6-s + 0.429·8-s + 0.333·9-s − 0.187·10-s + 0.732·11-s − 0.549·12-s − 0.587·13-s + 0.491·15-s + 0.857·16-s − 1.34·17-s − 0.0733·18-s + 1.72·19-s − 0.809·20-s − 0.161·22-s − 1.11·23-s + 0.247·24-s − 0.275·25-s + 0.129·26-s + 0.192·27-s − 0.852·29-s − 0.108·30-s − 0.521·31-s − 0.617·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 0.311T + 2T^{2} \) |
| 5 | \( 1 - 1.90T + 5T^{2} \) |
| 11 | \( 1 - 2.42T + 11T^{2} \) |
| 13 | \( 1 + 2.11T + 13T^{2} \) |
| 17 | \( 1 + 5.52T + 17T^{2} \) |
| 19 | \( 1 - 7.52T + 19T^{2} \) |
| 23 | \( 1 + 5.36T + 23T^{2} \) |
| 29 | \( 1 + 4.59T + 29T^{2} \) |
| 31 | \( 1 + 2.90T + 31T^{2} \) |
| 37 | \( 1 + 7.90T + 37T^{2} \) |
| 43 | \( 1 + 12.7T + 43T^{2} \) |
| 47 | \( 1 + 2.52T + 47T^{2} \) |
| 53 | \( 1 - 4.11T + 53T^{2} \) |
| 59 | \( 1 + 8.51T + 59T^{2} \) |
| 61 | \( 1 + 0.474T + 61T^{2} \) |
| 67 | \( 1 - 6.76T + 67T^{2} \) |
| 71 | \( 1 + 9.46T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 - 1.08T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 6.10T + 89T^{2} \) |
| 97 | \( 1 - 2.59T + 97T^{2} \) |
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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
−7.84083259202069988570998024748, −7.12133269468499688880442133202, −6.34900601323623457698744116641, −5.41413271462386523130379767220, −4.90910829429870119377985987092, −3.94035349138172490826626842349, −3.35989743708881068581872143405, −2.11967386706248624312573411130, −1.48208005743074995308092464660, 0,
1.48208005743074995308092464660, 2.11967386706248624312573411130, 3.35989743708881068581872143405, 3.94035349138172490826626842349, 4.90910829429870119377985987092, 5.41413271462386523130379767220, 6.34900601323623457698744116641, 7.12133269468499688880442133202, 7.84083259202069988570998024748
