| L(s) = 1 | + 5.72·2-s + 9·3-s + 0.820·4-s + 51.5·6-s − 158.·7-s − 178.·8-s + 81·9-s + 121·11-s + 7.38·12-s − 78.2·13-s − 906.·14-s − 1.04e3·16-s + 88.2·17-s + 464.·18-s + 1.86e3·19-s − 1.42e3·21-s + 693.·22-s − 503.·23-s − 1.60e3·24-s − 448.·26-s + 729·27-s − 129.·28-s + 1.05e3·29-s − 9.05e3·31-s − 297.·32-s + 1.08e3·33-s + 505.·34-s + ⋯ |
| L(s) = 1 | + 1.01·2-s + 0.577·3-s + 0.0256·4-s + 0.584·6-s − 1.22·7-s − 0.986·8-s + 0.333·9-s + 0.301·11-s + 0.0148·12-s − 0.128·13-s − 1.23·14-s − 1.02·16-s + 0.0740·17-s + 0.337·18-s + 1.18·19-s − 0.704·21-s + 0.305·22-s − 0.198·23-s − 0.569·24-s − 0.130·26-s + 0.192·27-s − 0.0312·28-s + 0.231·29-s − 1.69·31-s − 0.0512·32-s + 0.174·33-s + 0.0749·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.118849322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.118849322\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 5.72T + 32T^{2} \) |
| 7 | \( 1 + 158.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 78.2T + 3.71e5T^{2} \) |
| 17 | \( 1 - 88.2T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.86e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 503.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.05e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.05e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.89e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.45e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.91e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.43e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.55e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 9.90e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.56e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.68e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.14e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.57e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.04e5T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.54e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.77e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.25e5T + 8.58e9T^{2} \) |
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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
−9.400970610244730495437450831626, −8.867236115068984078124959773078, −7.61698627280072763631274348665, −6.72810397599502740533661139627, −5.86127586986834007868105521828, −4.98976177650669483066642870048, −3.77526113669909863278029837278, −3.37245029224776551306579407428, −2.31629596320909887117205713364, −0.64924553301945875929470892059,
0.64924553301945875929470892059, 2.31629596320909887117205713364, 3.37245029224776551306579407428, 3.77526113669909863278029837278, 4.98976177650669483066642870048, 5.86127586986834007868105521828, 6.72810397599502740533661139627, 7.61698627280072763631274348665, 8.867236115068984078124959773078, 9.400970610244730495437450831626
