| L(s) = 1 | + 6.72·2-s + 3.49·3-s + 13.1·4-s + 25·5-s + 23.4·6-s + 48.1·7-s − 126.·8-s − 230.·9-s + 168.·10-s + 121·11-s + 46.0·12-s + 1.09e3·13-s + 323.·14-s + 87.3·15-s − 1.27e3·16-s − 1.25e3·17-s − 1.55e3·18-s − 361·19-s + 329.·20-s + 168.·21-s + 813.·22-s − 471.·23-s − 442.·24-s + 625·25-s + 7.34e3·26-s − 1.65e3·27-s + 634.·28-s + ⋯ |
| L(s) = 1 | + 1.18·2-s + 0.224·3-s + 0.411·4-s + 0.447·5-s + 0.266·6-s + 0.371·7-s − 0.698·8-s − 0.949·9-s + 0.531·10-s + 0.301·11-s + 0.0923·12-s + 1.79·13-s + 0.440·14-s + 0.100·15-s − 1.24·16-s − 1.05·17-s − 1.12·18-s − 0.229·19-s + 0.184·20-s + 0.0832·21-s + 0.358·22-s − 0.185·23-s − 0.156·24-s + 0.200·25-s + 2.13·26-s − 0.437·27-s + 0.152·28-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 | \( 1 - 6.72T + 32T^{2} \) |
| 3 | \( 1 - 3.49T + 243T^{2} \) |
| 7 | \( 1 - 48.1T + 1.68e4T^{2} \) |
| 13 | \( 1 - 1.09e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.25e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 471.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.96e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.63e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.86e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.52e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 3.14e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.94e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 6.92e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.89e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.53e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.72e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.64e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.03e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.85e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.14e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.02e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.44e4T + 8.58e9T^{2} \) |
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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
−8.753380914334292293770688814705, −8.130722047327605368755104433349, −6.58807593354103511060693164709, −6.10594504083271662978644238037, −5.32662573326790565714086951631, −4.33223149049205205190728745282, −3.55640177231178808213966855325, −2.63593940722763365282635288964, −1.52446324156223638479268671315, 0,
1.52446324156223638479268671315, 2.63593940722763365282635288964, 3.55640177231178808213966855325, 4.33223149049205205190728745282, 5.32662573326790565714086951631, 6.10594504083271662978644238037, 6.58807593354103511060693164709, 8.130722047327605368755104433349, 8.753380914334292293770688814705
