| L(s) = 1 | − 9.14·2-s + 9·3-s + 51.6·4-s − 25·5-s − 82.3·6-s + 24.2·7-s − 180.·8-s + 81·9-s + 228.·10-s − 126.·11-s + 465.·12-s + 169·13-s − 221.·14-s − 225·15-s − 6.48·16-s − 912.·17-s − 740.·18-s + 507.·19-s − 1.29e3·20-s + 217.·21-s + 1.15e3·22-s − 1.47e3·23-s − 1.62e3·24-s + 625·25-s − 1.54e3·26-s + 729·27-s + 1.25e3·28-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 0.577·3-s + 1.61·4-s − 0.447·5-s − 0.933·6-s + 0.186·7-s − 0.994·8-s + 0.333·9-s + 0.723·10-s − 0.315·11-s + 0.932·12-s + 0.277·13-s − 0.301·14-s − 0.258·15-s − 0.00633·16-s − 0.766·17-s − 0.539·18-s + 0.322·19-s − 0.722·20-s + 0.107·21-s + 0.509·22-s − 0.581·23-s − 0.574·24-s + 0.200·25-s − 0.448·26-s + 0.192·27-s + 0.301·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 9T \) |
| 5 | \( 1 + 25T \) |
| 13 | \( 1 - 169T \) |
| good | 2 | \( 1 + 9.14T + 32T^{2} \) |
| 7 | \( 1 - 24.2T + 1.68e4T^{2} \) |
| 11 | \( 1 + 126.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 912.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 507.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.47e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 315.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.16e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.46e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.92e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.15e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.33e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.54e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.23e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.87e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.11e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.82e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 9.88e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.96e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.67e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.83e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.05e4T + 8.58e9T^{2} \) |
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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
−10.89225509197496627266135567447, −9.910348416505816484392011584798, −9.056503678323827864141931705861, −8.162551415726364481742470069843, −7.56024605628931994926660008009, −6.37099027776451040979764123981, −4.44424678719721967154390018106, −2.78228297888674052298435819864, −1.44711984119799416649266218933, 0,
1.44711984119799416649266218933, 2.78228297888674052298435819864, 4.44424678719721967154390018106, 6.37099027776451040979764123981, 7.56024605628931994926660008009, 8.162551415726364481742470069843, 9.056503678323827864141931705861, 9.910348416505816484392011584798, 10.89225509197496627266135567447
