| L(s) = 1 | + 1.85·2-s + 1.37·3-s + 1.45·4-s − 5-s + 2.54·6-s − 7-s − 1.00·8-s − 1.12·9-s − 1.85·10-s + 0.632·11-s + 1.99·12-s + 3.17·13-s − 1.85·14-s − 1.37·15-s − 4.79·16-s + 1.94·17-s − 2.08·18-s + 5.06·19-s − 1.45·20-s − 1.37·21-s + 1.17·22-s − 9.20·23-s − 1.38·24-s + 25-s + 5.91·26-s − 5.64·27-s − 1.45·28-s + ⋯ |
| L(s) = 1 | + 1.31·2-s + 0.791·3-s + 0.728·4-s − 0.447·5-s + 1.04·6-s − 0.377·7-s − 0.356·8-s − 0.374·9-s − 0.587·10-s + 0.190·11-s + 0.576·12-s + 0.881·13-s − 0.496·14-s − 0.353·15-s − 1.19·16-s + 0.470·17-s − 0.491·18-s + 1.16·19-s − 0.325·20-s − 0.299·21-s + 0.250·22-s − 1.91·23-s − 0.282·24-s + 0.200·25-s + 1.15·26-s − 1.08·27-s − 0.275·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 - 1.85T + 2T^{2} \) |
| 3 | \( 1 - 1.37T + 3T^{2} \) |
| 11 | \( 1 - 0.632T + 11T^{2} \) |
| 13 | \( 1 - 3.17T + 13T^{2} \) |
| 17 | \( 1 - 1.94T + 17T^{2} \) |
| 19 | \( 1 - 5.06T + 19T^{2} \) |
| 23 | \( 1 + 9.20T + 23T^{2} \) |
| 29 | \( 1 + 2.38T + 29T^{2} \) |
| 31 | \( 1 - 2.40T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 1.10T + 41T^{2} \) |
| 43 | \( 1 - 5.93T + 43T^{2} \) |
| 47 | \( 1 - 1.30T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 7.04T + 59T^{2} \) |
| 61 | \( 1 - 2.94T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 0.265T + 71T^{2} \) |
| 73 | \( 1 + 9.57T + 73T^{2} \) |
| 79 | \( 1 - 2.02T + 79T^{2} \) |
| 83 | \( 1 + 4.07T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 6.49T + 97T^{2} \) |
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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
−7.56010261955596418524388847440, −6.56628808061348877131113023221, −5.91737442975314894588342173769, −5.45140085877170879649236396772, −4.45652432056577991444869490142, −3.73527817181321394873974947024, −3.36482357050101796757331898463, −2.69629293730318540769273757834, −1.60981666757869286888123083288, 0,
1.60981666757869286888123083288, 2.69629293730318540769273757834, 3.36482357050101796757331898463, 3.73527817181321394873974947024, 4.45652432056577991444869490142, 5.45140085877170879649236396772, 5.91737442975314894588342173769, 6.56628808061348877131113023221, 7.56010261955596418524388847440
