| L(s) = 1 | − 10.1·2-s − 9·3-s + 71.8·4-s + 91.7·6-s − 134.·7-s − 405.·8-s + 81·9-s − 121·11-s − 646.·12-s − 328.·13-s + 1.37e3·14-s + 1.83e3·16-s − 636.·17-s − 825.·18-s − 975.·19-s + 1.21e3·21-s + 1.23e3·22-s + 1.14e3·23-s + 3.65e3·24-s + 3.34e3·26-s − 729·27-s − 9.67e3·28-s − 1.07e3·29-s + 8.56e3·31-s − 5.73e3·32-s + 1.08e3·33-s + 6.48e3·34-s + ⋯ |
| L(s) = 1 | − 1.80·2-s − 0.577·3-s + 2.24·4-s + 1.04·6-s − 1.03·7-s − 2.24·8-s + 0.333·9-s − 0.301·11-s − 1.29·12-s − 0.539·13-s + 1.87·14-s + 1.79·16-s − 0.534·17-s − 0.600·18-s − 0.620·19-s + 0.599·21-s + 0.543·22-s + 0.450·23-s + 1.29·24-s + 0.971·26-s − 0.192·27-s − 2.33·28-s − 0.236·29-s + 1.60·31-s − 0.990·32-s + 0.174·33-s + 0.962·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 + 10.1T + 32T^{2} \) |
| 7 | \( 1 + 134.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 328.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 636.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 975.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.14e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.07e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.56e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.88e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.04e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.49e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.53e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 5.13e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.36e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.57e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.12e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.28e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.21e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.26e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.01e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.67e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.80e5T + 8.58e9T^{2} \) |
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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
−9.190787733997555638335529775170, −8.329763786067170523385066211580, −7.42970644923007156011937821411, −6.63505102248489861982301898867, −6.08444902625835295384385148264, −4.65365677954389179906365852395, −3.06756148598117540595030914481, −2.09790114864736920538608923887, −0.789318143404597858223279793765, 0,
0.789318143404597858223279793765, 2.09790114864736920538608923887, 3.06756148598117540595030914481, 4.65365677954389179906365852395, 6.08444902625835295384385148264, 6.63505102248489861982301898867, 7.42970644923007156011937821411, 8.329763786067170523385066211580, 9.190787733997555638335529775170
