| L(s) = 1 | + 2.08·2-s + 3-s + 2.32·4-s − 5-s + 2.08·6-s − 4.12·7-s + 0.680·8-s + 9-s − 2.08·10-s + 0.803·11-s + 2.32·12-s − 3.58·13-s − 8.57·14-s − 15-s − 3.23·16-s − 4.24·17-s + 2.08·18-s − 2.93·19-s − 2.32·20-s − 4.12·21-s + 1.67·22-s + 8.73·23-s + 0.680·24-s + 25-s − 7.46·26-s + 27-s − 9.58·28-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 0.577·3-s + 1.16·4-s − 0.447·5-s + 0.849·6-s − 1.55·7-s + 0.240·8-s + 0.333·9-s − 0.657·10-s + 0.242·11-s + 0.671·12-s − 0.994·13-s − 2.29·14-s − 0.258·15-s − 0.809·16-s − 1.03·17-s + 0.490·18-s − 0.672·19-s − 0.520·20-s − 0.899·21-s + 0.356·22-s + 1.82·23-s + 0.138·24-s + 0.200·25-s − 1.46·26-s + 0.192·27-s − 1.81·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 163 | \( 1 - T \) |
| good | 2 | \( 1 - 2.08T + 2T^{2} \) |
| 7 | \( 1 + 4.12T + 7T^{2} \) |
| 11 | \( 1 - 0.803T + 11T^{2} \) |
| 13 | \( 1 + 3.58T + 13T^{2} \) |
| 17 | \( 1 + 4.24T + 17T^{2} \) |
| 19 | \( 1 + 2.93T + 19T^{2} \) |
| 23 | \( 1 - 8.73T + 23T^{2} \) |
| 29 | \( 1 - 1.04T + 29T^{2} \) |
| 31 | \( 1 + 0.578T + 31T^{2} \) |
| 37 | \( 1 + 7.28T + 37T^{2} \) |
| 41 | \( 1 + 8.20T + 41T^{2} \) |
| 43 | \( 1 + 8.86T + 43T^{2} \) |
| 47 | \( 1 - 3.41T + 47T^{2} \) |
| 53 | \( 1 - 0.191T + 53T^{2} \) |
| 59 | \( 1 + 7.66T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 - 0.177T + 67T^{2} \) |
| 71 | \( 1 - 8.02T + 71T^{2} \) |
| 73 | \( 1 - 3.89T + 73T^{2} \) |
| 79 | \( 1 - 4.26T + 79T^{2} \) |
| 83 | \( 1 - 6.03T + 83T^{2} \) |
| 89 | \( 1 - 3.37T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−8.802735813500576689700060008757, −7.43048999253574531344314546377, −6.71013991600190812145841228040, −6.40658501740859662966140529325, −5.09975553388073672561447770522, −4.55728489596418939634596332749, −3.49832423615567021893694086300, −3.14372017989644283499245768625, −2.18896185571363472583144856218, 0,
2.18896185571363472583144856218, 3.14372017989644283499245768625, 3.49832423615567021893694086300, 4.55728489596418939634596332749, 5.09975553388073672561447770522, 6.40658501740859662966140529325, 6.71013991600190812145841228040, 7.43048999253574531344314546377, 8.802735813500576689700060008757
