| L(s) = 1 | − 1.30·2-s + 1.46·3-s − 0.284·4-s − 5-s − 1.92·6-s + 3.81·7-s + 2.99·8-s − 0.841·9-s + 1.30·10-s + 0.423·11-s − 0.418·12-s − 1.12·13-s − 5.00·14-s − 1.46·15-s − 3.34·16-s − 1.97·17-s + 1.10·18-s − 6.29·19-s + 0.284·20-s + 5.61·21-s − 0.554·22-s + 4.39·24-s + 25-s + 1.47·26-s − 5.64·27-s − 1.08·28-s + 3.09·29-s + ⋯ |
| L(s) = 1 | − 0.926·2-s + 0.848·3-s − 0.142·4-s − 0.447·5-s − 0.785·6-s + 1.44·7-s + 1.05·8-s − 0.280·9-s + 0.414·10-s + 0.127·11-s − 0.120·12-s − 0.312·13-s − 1.33·14-s − 0.379·15-s − 0.837·16-s − 0.478·17-s + 0.259·18-s − 1.44·19-s + 0.0636·20-s + 1.22·21-s − 0.118·22-s + 0.897·24-s + 0.200·25-s + 0.289·26-s − 1.08·27-s − 0.205·28-s + 0.574·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 1.30T + 2T^{2} \) |
| 3 | \( 1 - 1.46T + 3T^{2} \) |
| 7 | \( 1 - 3.81T + 7T^{2} \) |
| 11 | \( 1 - 0.423T + 11T^{2} \) |
| 13 | \( 1 + 1.12T + 13T^{2} \) |
| 17 | \( 1 + 1.97T + 17T^{2} \) |
| 19 | \( 1 + 6.29T + 19T^{2} \) |
| 29 | \( 1 - 3.09T + 29T^{2} \) |
| 31 | \( 1 + 3.32T + 31T^{2} \) |
| 37 | \( 1 + 3.06T + 37T^{2} \) |
| 41 | \( 1 - 5.81T + 41T^{2} \) |
| 43 | \( 1 + 7.91T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 4.27T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 7.46T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + 9.37T + 73T^{2} \) |
| 79 | \( 1 - 4.13T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 - 3.92T + 89T^{2} \) |
| 97 | \( 1 - 3.59T + 97T^{2} \) |
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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
−8.500091426713539024636768480338, −8.028224795744745452343695659107, −7.44451629692448138105497138689, −6.42667251159245826565632871216, −5.06555116187795998921660641356, −4.53020222469516778653120484297, −3.63056680667628013711249834545, −2.31933373826851762469943961954, −1.56065119985604737250943776280, 0,
1.56065119985604737250943776280, 2.31933373826851762469943961954, 3.63056680667628013711249834545, 4.53020222469516778653120484297, 5.06555116187795998921660641356, 6.42667251159245826565632871216, 7.44451629692448138105497138689, 8.028224795744745452343695659107, 8.500091426713539024636768480338
