| L(s) = 1 | − 2.25·2-s − 0.982·3-s + 3.08·4-s + 2.20·5-s + 2.21·6-s + 0.164·7-s − 2.44·8-s − 2.03·9-s − 4.96·10-s − 1.46·11-s − 3.02·12-s − 5.51·13-s − 0.371·14-s − 2.16·15-s − 0.656·16-s − 5.86·17-s + 4.58·18-s + 2.55·19-s + 6.79·20-s − 0.161·21-s + 3.31·22-s + 4.06·23-s + 2.40·24-s − 0.146·25-s + 12.4·26-s + 4.94·27-s + 0.507·28-s + ⋯ |
| L(s) = 1 | − 1.59·2-s − 0.567·3-s + 1.54·4-s + 0.985·5-s + 0.904·6-s + 0.0622·7-s − 0.864·8-s − 0.678·9-s − 1.57·10-s − 0.442·11-s − 0.874·12-s − 1.53·13-s − 0.0992·14-s − 0.558·15-s − 0.164·16-s − 1.42·17-s + 1.08·18-s + 0.585·19-s + 1.51·20-s − 0.0353·21-s + 0.705·22-s + 0.848·23-s + 0.490·24-s − 0.0293·25-s + 2.44·26-s + 0.951·27-s + 0.0959·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 37 | \( 1 + T \) |
| 109 | \( 1 + T \) |
| good | 2 | \( 1 + 2.25T + 2T^{2} \) |
| 3 | \( 1 + 0.982T + 3T^{2} \) |
| 5 | \( 1 - 2.20T + 5T^{2} \) |
| 7 | \( 1 - 0.164T + 7T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 13 | \( 1 + 5.51T + 13T^{2} \) |
| 17 | \( 1 + 5.86T + 17T^{2} \) |
| 19 | \( 1 - 2.55T + 19T^{2} \) |
| 23 | \( 1 - 4.06T + 23T^{2} \) |
| 29 | \( 1 - 5.77T + 29T^{2} \) |
| 31 | \( 1 - 7.63T + 31T^{2} \) |
| 41 | \( 1 - 7.80T + 41T^{2} \) |
| 43 | \( 1 - 8.43T + 43T^{2} \) |
| 47 | \( 1 + 0.369T + 47T^{2} \) |
| 53 | \( 1 + 8.21T + 53T^{2} \) |
| 59 | \( 1 - 9.69T + 59T^{2} \) |
| 61 | \( 1 - 3.52T + 61T^{2} \) |
| 67 | \( 1 - 3.90T + 67T^{2} \) |
| 71 | \( 1 + 5.46T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 - 2.31T + 79T^{2} \) |
| 83 | \( 1 + 7.50T + 83T^{2} \) |
| 89 | \( 1 + 1.80T + 89T^{2} \) |
| 97 | \( 1 - 5.91T + 97T^{2} \) |
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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
−8.235877747522674056460609914220, −7.44941750951991601744644801059, −6.71338456468773099411945488425, −6.15017027726252895643701696876, −5.17163843385248938819844820438, −4.60346713228926878029107820538, −2.62035898896556869733077762557, −2.44125894532596691918752985710, −1.07745066966431521612785244579, 0,
1.07745066966431521612785244579, 2.44125894532596691918752985710, 2.62035898896556869733077762557, 4.60346713228926878029107820538, 5.17163843385248938819844820438, 6.15017027726252895643701696876, 6.71338456468773099411945488425, 7.44941750951991601744644801059, 8.235877747522674056460609914220
