| L(s) = 1 | − 0.656·2-s + 2.76·3-s − 1.56·4-s + 5-s − 1.81·6-s + 7-s + 2.34·8-s + 4.64·9-s − 0.656·10-s − 5.20·11-s − 4.33·12-s − 0.887·13-s − 0.656·14-s + 2.76·15-s + 1.59·16-s − 3.19·17-s − 3.05·18-s − 3.11·19-s − 1.56·20-s + 2.76·21-s + 3.41·22-s − 3.41·23-s + 6.48·24-s + 25-s + 0.582·26-s + 4.56·27-s − 1.56·28-s + ⋯ |
| L(s) = 1 | − 0.464·2-s + 1.59·3-s − 0.784·4-s + 0.447·5-s − 0.741·6-s + 0.377·7-s + 0.828·8-s + 1.54·9-s − 0.207·10-s − 1.56·11-s − 1.25·12-s − 0.246·13-s − 0.175·14-s + 0.714·15-s + 0.399·16-s − 0.775·17-s − 0.719·18-s − 0.715·19-s − 0.350·20-s + 0.603·21-s + 0.728·22-s − 0.711·23-s + 1.32·24-s + 0.200·25-s + 0.114·26-s + 0.878·27-s − 0.296·28-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 | \( 1 + 0.656T + 2T^{2} \) |
| 3 | \( 1 - 2.76T + 3T^{2} \) |
| 11 | \( 1 + 5.20T + 11T^{2} \) |
| 13 | \( 1 + 0.887T + 13T^{2} \) |
| 17 | \( 1 + 3.19T + 17T^{2} \) |
| 19 | \( 1 + 3.11T + 19T^{2} \) |
| 23 | \( 1 + 3.41T + 23T^{2} \) |
| 29 | \( 1 + 0.110T + 29T^{2} \) |
| 31 | \( 1 - 3.91T + 31T^{2} \) |
| 37 | \( 1 - 8.91T + 37T^{2} \) |
| 41 | \( 1 - 3.61T + 41T^{2} \) |
| 43 | \( 1 + 2.94T + 43T^{2} \) |
| 47 | \( 1 + 2.00T + 47T^{2} \) |
| 53 | \( 1 - 6.16T + 53T^{2} \) |
| 59 | \( 1 + 4.92T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 0.0752T + 67T^{2} \) |
| 71 | \( 1 - 5.92T + 71T^{2} \) |
| 73 | \( 1 - 3.40T + 73T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 + 18.1T + 89T^{2} \) |
| 97 | \( 1 + 3.53T + 97T^{2} \) |
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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
−7.85035015726766058216796398901, −7.20932950445070779164931990802, −6.13544917215413292141710902891, −5.21558544137995809417542062080, −4.49509032476920202629155669784, −3.96273852779034043879474316965, −2.77606186674634186904920546970, −2.38945908704297926337594552006, −1.44958262905527360648299669577, 0,
1.44958262905527360648299669577, 2.38945908704297926337594552006, 2.77606186674634186904920546970, 3.96273852779034043879474316965, 4.49509032476920202629155669784, 5.21558544137995809417542062080, 6.13544917215413292141710902891, 7.20932950445070779164931990802, 7.85035015726766058216796398901
