| L(s) = 1 | + 3.20·2-s − 29.5·3-s − 21.7·4-s − 94.8·6-s − 11.5·7-s − 172.·8-s + 632.·9-s − 596.·11-s + 642.·12-s + 169·13-s − 37.1·14-s + 142.·16-s + 2.09e3·17-s + 2.02e3·18-s + 35.4·19-s + 342.·21-s − 1.91e3·22-s + 2.78e3·23-s + 5.09e3·24-s + 541.·26-s − 1.15e4·27-s + 251.·28-s − 370.·29-s + 5.05e3·31-s + 5.96e3·32-s + 1.76e4·33-s + 6.71e3·34-s + ⋯ |
| L(s) = 1 | + 0.566·2-s − 1.89·3-s − 0.678·4-s − 1.07·6-s − 0.0892·7-s − 0.951·8-s + 2.60·9-s − 1.48·11-s + 1.28·12-s + 0.277·13-s − 0.0506·14-s + 0.139·16-s + 1.75·17-s + 1.47·18-s + 0.0225·19-s + 0.169·21-s − 0.842·22-s + 1.09·23-s + 1.80·24-s + 0.157·26-s − 3.04·27-s + 0.0605·28-s − 0.0817·29-s + 0.944·31-s + 1.03·32-s + 2.82·33-s + 0.995·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - 169T \) |
| good | 2 | \( 1 - 3.20T + 32T^{2} \) |
| 3 | \( 1 + 29.5T + 243T^{2} \) |
| 7 | \( 1 + 11.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + 596.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 2.09e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 35.4T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.78e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 370.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.05e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.12e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.81e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.90e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.31e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.82e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.78e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 7.39e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.33e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.31e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.08e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.77e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.42e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.64e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.54e5T + 8.58e9T^{2} \) |
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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
−10.27662660418272348176641672690, −9.927017616147294011749118418224, −8.285430758176530522835944676595, −7.11302357847818106064733775524, −5.93049367935974027215997573311, −5.30231559823003218437995258480, −4.68655594195722039291424104924, −3.27903928286136950592885603599, −1.03803694548153880708349090320, 0,
1.03803694548153880708349090320, 3.27903928286136950592885603599, 4.68655594195722039291424104924, 5.30231559823003218437995258480, 5.93049367935974027215997573311, 7.11302357847818106064733775524, 8.285430758176530522835944676595, 9.927017616147294011749118418224, 10.27662660418272348176641672690
