| L(s) = 1 | − 3.41·2-s − 5.73·3-s − 20.3·4-s − 25·5-s + 19.6·6-s + 236.·7-s + 178.·8-s − 210.·9-s + 85.4·10-s + 121·11-s + 116.·12-s + 1.09e3·13-s − 806.·14-s + 143.·15-s + 39.0·16-s − 1.68e3·17-s + 718.·18-s + 361·19-s + 507.·20-s − 1.35e3·21-s − 413.·22-s + 572.·23-s − 1.02e3·24-s + 625·25-s − 3.75e3·26-s + 2.59e3·27-s − 4.79e3·28-s + ⋯ |
| L(s) = 1 | − 0.604·2-s − 0.367·3-s − 0.634·4-s − 0.447·5-s + 0.222·6-s + 1.82·7-s + 0.987·8-s − 0.864·9-s + 0.270·10-s + 0.301·11-s + 0.233·12-s + 1.80·13-s − 1.10·14-s + 0.164·15-s + 0.0381·16-s − 1.41·17-s + 0.522·18-s + 0.229·19-s + 0.283·20-s − 0.670·21-s − 0.182·22-s + 0.225·23-s − 0.363·24-s + 0.200·25-s − 1.08·26-s + 0.686·27-s − 1.15·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 3.41T + 32T^{2} \) |
| 3 | \( 1 + 5.73T + 243T^{2} \) |
| 7 | \( 1 - 236.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 1.09e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.68e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 572.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.15e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.93e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 517.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.33e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 236.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.22e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.03e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.62e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.76e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.16e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.97e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.33e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.41e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.01e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.05e5T + 8.58e9T^{2} \) |
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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
−8.831192294164573816837033512921, −8.107676467649452671609169575283, −7.46300143823600533968334886201, −6.13838383599708508715840151103, −5.24843296103629591791257541601, −4.44703609436801599423266567387, −3.65252098441052138718653073481, −1.90043955935239697554837739339, −1.08387824538024811758588747025, 0,
1.08387824538024811758588747025, 1.90043955935239697554837739339, 3.65252098441052138718653073481, 4.44703609436801599423266567387, 5.24843296103629591791257541601, 6.13838383599708508715840151103, 7.46300143823600533968334886201, 8.107676467649452671609169575283, 8.831192294164573816837033512921
