| L(s) = 1 | − 2.67·2-s − 11.1·3-s − 24.8·4-s + 25·5-s + 29.7·6-s + 158.·7-s + 151.·8-s − 119.·9-s − 66.8·10-s + 121·11-s + 276.·12-s + 28.0·13-s − 424.·14-s − 278.·15-s + 388.·16-s + 594.·17-s + 318.·18-s − 361·19-s − 621.·20-s − 1.76e3·21-s − 323.·22-s − 639.·23-s − 1.69e3·24-s + 625·25-s − 75.1·26-s + 4.03e3·27-s − 3.94e3·28-s + ⋯ |
| L(s) = 1 | − 0.472·2-s − 0.714·3-s − 0.776·4-s + 0.447·5-s + 0.337·6-s + 1.22·7-s + 0.839·8-s − 0.490·9-s − 0.211·10-s + 0.301·11-s + 0.554·12-s + 0.0461·13-s − 0.579·14-s − 0.319·15-s + 0.379·16-s + 0.499·17-s + 0.231·18-s − 0.229·19-s − 0.347·20-s − 0.875·21-s − 0.142·22-s − 0.251·23-s − 0.599·24-s + 0.200·25-s − 0.0217·26-s + 1.06·27-s − 0.951·28-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 | \( 1 + 2.67T + 32T^{2} \) |
| 3 | \( 1 + 11.1T + 243T^{2} \) |
| 7 | \( 1 - 158.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 28.0T + 3.71e5T^{2} \) |
| 17 | \( 1 - 594.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 639.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.51e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.79e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.48e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.76e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.61e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.81e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.00e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.83e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.95e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.29e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.94e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.16e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.22e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.56e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 570.T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.77e5T + 8.58e9T^{2} \) |
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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
−8.729523573155949931578702702692, −8.170087203102391649322155292151, −7.21915284716735108639020262239, −6.05716999636026521099627434978, −5.23461019906784696950597631403, −4.72033297648406893085865603822, −3.52715467290135538291256934214, −1.93972849025771988703805673517, −1.06680627603823940584757622315, 0,
1.06680627603823940584757622315, 1.93972849025771988703805673517, 3.52715467290135538291256934214, 4.72033297648406893085865603822, 5.23461019906784696950597631403, 6.05716999636026521099627434978, 7.21915284716735108639020262239, 8.170087203102391649322155292151, 8.729523573155949931578702702692
