| L(s) = 1 | + 2-s + 4-s + 0.181·5-s + 3.43·7-s + 8-s + 0.181·10-s − 5.90·11-s − 5.08·13-s + 3.43·14-s + 16-s + 1.36·17-s − 0.833·19-s + 0.181·20-s − 5.90·22-s − 4.39·23-s − 4.96·25-s − 5.08·26-s + 3.43·28-s + 9.69·29-s − 2.47·31-s + 32-s + 1.36·34-s + 0.622·35-s + 10.6·37-s − 0.833·38-s + 0.181·40-s + 1.92·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.0810·5-s + 1.29·7-s + 0.353·8-s + 0.0573·10-s − 1.77·11-s − 1.41·13-s + 0.918·14-s + 0.250·16-s + 0.330·17-s − 0.191·19-s + 0.0405·20-s − 1.25·22-s − 0.915·23-s − 0.993·25-s − 0.997·26-s + 0.649·28-s + 1.79·29-s − 0.444·31-s + 0.176·32-s + 0.234·34-s + 0.105·35-s + 1.75·37-s − 0.135·38-s + 0.0286·40-s + 0.301·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 - 0.181T + 5T^{2} \) |
| 7 | \( 1 - 3.43T + 7T^{2} \) |
| 11 | \( 1 + 5.90T + 11T^{2} \) |
| 13 | \( 1 + 5.08T + 13T^{2} \) |
| 17 | \( 1 - 1.36T + 17T^{2} \) |
| 19 | \( 1 + 0.833T + 19T^{2} \) |
| 23 | \( 1 + 4.39T + 23T^{2} \) |
| 29 | \( 1 - 9.69T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 - 1.92T + 41T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 + 3.91T + 47T^{2} \) |
| 53 | \( 1 - 2.25T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 + 9.69T + 61T^{2} \) |
| 67 | \( 1 + 8.45T + 67T^{2} \) |
| 71 | \( 1 + 5.45T + 71T^{2} \) |
| 73 | \( 1 + 0.635T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 + 0.241T + 83T^{2} \) |
| 89 | \( 1 - 2.03T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−7.66390911611021867866925108897, −6.81116053769548979601328227807, −5.82138535649666309733014879103, −5.28926561679849153077743912750, −4.71735731486027892791503264696, −4.19966865208883914166710822893, −2.87729104964500984998550256708, −2.43813670173509841599752120866, −1.55374227226669598553487297063, 0,
1.55374227226669598553487297063, 2.43813670173509841599752120866, 2.87729104964500984998550256708, 4.19966865208883914166710822893, 4.71735731486027892791503264696, 5.28926561679849153077743912750, 5.82138535649666309733014879103, 6.81116053769548979601328227807, 7.66390911611021867866925108897
