| L(s) = 1 | − 2.62·2-s + 3-s + 4.90·4-s + 2.26·5-s − 2.62·6-s − 0.677·7-s − 7.64·8-s + 9-s − 5.96·10-s + 11-s + 4.90·12-s − 6.23·13-s + 1.78·14-s + 2.26·15-s + 10.2·16-s − 4.02·17-s − 2.62·18-s + 5.38·19-s + 11.1·20-s − 0.677·21-s − 2.62·22-s − 7.11·23-s − 7.64·24-s + 0.144·25-s + 16.3·26-s + 27-s − 3.32·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.577·3-s + 2.45·4-s + 1.01·5-s − 1.07·6-s − 0.256·7-s − 2.70·8-s + 0.333·9-s − 1.88·10-s + 0.301·11-s + 1.41·12-s − 1.72·13-s + 0.476·14-s + 0.585·15-s + 2.56·16-s − 0.975·17-s − 0.619·18-s + 1.23·19-s + 2.48·20-s − 0.147·21-s − 0.560·22-s − 1.48·23-s − 1.55·24-s + 0.0289·25-s + 3.21·26-s + 0.192·27-s − 0.628·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 5 | \( 1 - 2.26T + 5T^{2} \) |
| 7 | \( 1 + 0.677T + 7T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 + 4.02T + 17T^{2} \) |
| 19 | \( 1 - 5.38T + 19T^{2} \) |
| 23 | \( 1 + 7.11T + 23T^{2} \) |
| 29 | \( 1 + 9.43T + 29T^{2} \) |
| 31 | \( 1 - 8.36T + 31T^{2} \) |
| 37 | \( 1 - 1.53T + 37T^{2} \) |
| 41 | \( 1 - 2.74T + 41T^{2} \) |
| 43 | \( 1 + 8.78T + 43T^{2} \) |
| 47 | \( 1 + 4.57T + 47T^{2} \) |
| 53 | \( 1 + 8.79T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 67 | \( 1 + 0.817T + 67T^{2} \) |
| 71 | \( 1 + 8.00T + 71T^{2} \) |
| 73 | \( 1 + 5.45T + 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 - 8.66T + 83T^{2} \) |
| 89 | \( 1 - 3.13T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−9.076240359322204363326947072431, −7.941172117770506446772499532542, −7.60381648727054867467306233800, −6.64378238339986069258788680973, −6.04668104876533719002755616572, −4.79596034927934424438315063766, −3.18904476616978884264392879235, −2.24902200358857907426861098263, −1.67378125177348337442071963871, 0,
1.67378125177348337442071963871, 2.24902200358857907426861098263, 3.18904476616978884264392879235, 4.79596034927934424438315063766, 6.04668104876533719002755616572, 6.64378238339986069258788680973, 7.60381648727054867467306233800, 7.941172117770506446772499532542, 9.076240359322204363326947072431
