| L(s) = 1 | − 10.2·2-s + 5.52·3-s + 73.3·4-s − 56.6·6-s − 68.9·7-s − 423.·8-s − 212.·9-s − 486.·11-s + 404.·12-s + 428.·13-s + 707.·14-s + 2.00e3·16-s − 1.80e3·17-s + 2.18e3·18-s − 1.04e3·19-s − 380.·21-s + 4.98e3·22-s − 686.·23-s − 2.34e3·24-s − 4.39e3·26-s − 2.51e3·27-s − 5.05e3·28-s − 1.33e3·29-s + 7.99e3·31-s − 7.00e3·32-s − 2.68e3·33-s + 1.84e4·34-s + ⋯ |
| L(s) = 1 | − 1.81·2-s + 0.354·3-s + 2.29·4-s − 0.642·6-s − 0.531·7-s − 2.34·8-s − 0.874·9-s − 1.21·11-s + 0.811·12-s + 0.703·13-s + 0.964·14-s + 1.95·16-s − 1.51·17-s + 1.58·18-s − 0.665·19-s − 0.188·21-s + 2.19·22-s − 0.270·23-s − 0.829·24-s − 1.27·26-s − 0.664·27-s − 1.21·28-s − 0.295·29-s + 1.49·31-s − 1.20·32-s − 0.429·33-s + 2.74·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 10.2T + 32T^{2} \) |
| 3 | \( 1 - 5.52T + 243T^{2} \) |
| 7 | \( 1 + 68.9T + 1.68e4T^{2} \) |
| 11 | \( 1 + 486.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 428.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.80e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.04e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 686.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.33e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.99e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.97e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.07e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.50e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 895.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.93e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.11e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.77e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 7.71e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.14e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.37e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.22e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.29e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.46e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.48e5T + 8.58e9T^{2} \) |
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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
−16.16592811903798835547710940696, −15.28662053889964568448422871021, −13.30109137905299356862888136798, −11.39091308761606941786167829024, −10.33507100532086477909404827690, −8.948011093739175151638306240357, −8.057820782844069705287439738233, −6.38780778736165990773067510705, −2.52685205639872386856587465629, 0,
2.52685205639872386856587465629, 6.38780778736165990773067510705, 8.057820782844069705287439738233, 8.948011093739175151638306240357, 10.33507100532086477909404827690, 11.39091308761606941786167829024, 13.30109137905299356862888136798, 15.28662053889964568448422871021, 16.16592811903798835547710940696
