| L(s) = 1 | + 6.43·2-s − 9·3-s + 9.39·4-s − 57.9·6-s − 9.96·7-s − 145.·8-s + 81·9-s − 121·11-s − 84.5·12-s + 813.·13-s − 64.1·14-s − 1.23e3·16-s − 616.·17-s + 521.·18-s + 1.98e3·19-s + 89.7·21-s − 778.·22-s − 545.·23-s + 1.30e3·24-s + 5.23e3·26-s − 729·27-s − 93.6·28-s − 670.·29-s − 1.22e3·31-s − 3.30e3·32-s + 1.08e3·33-s − 3.96e3·34-s + ⋯ |
| L(s) = 1 | + 1.13·2-s − 0.577·3-s + 0.293·4-s − 0.656·6-s − 0.0768·7-s − 0.803·8-s + 0.333·9-s − 0.301·11-s − 0.169·12-s + 1.33·13-s − 0.0874·14-s − 1.20·16-s − 0.516·17-s + 0.379·18-s + 1.26·19-s + 0.0443·21-s − 0.342·22-s − 0.215·23-s + 0.463·24-s + 1.51·26-s − 0.192·27-s − 0.0225·28-s − 0.148·29-s − 0.229·31-s − 0.569·32-s + 0.174·33-s − 0.588·34-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 | \( 1 - 6.43T + 32T^{2} \) |
| 7 | \( 1 + 9.96T + 1.68e4T^{2} \) |
| 13 | \( 1 - 813.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 616.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.98e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 545.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 670.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.22e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.82e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.58e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.30e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.06e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.18e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.15e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 6.74e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.89e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.52e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.39e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.01e5T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.45e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.04e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.86e4T + 8.58e9T^{2} \) |
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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
−9.115342582428902571028420820495, −8.151065434349729714066319725865, −6.99239348446402368047701012396, −6.08485689166702535778556694051, −5.52934932335464165212039498981, −4.58002134868003337912326046080, −3.75280783457101324887521138066, −2.80895281050141940180170486786, −1.29035994171168230125239946704, 0,
1.29035994171168230125239946704, 2.80895281050141940180170486786, 3.75280783457101324887521138066, 4.58002134868003337912326046080, 5.52934932335464165212039498981, 6.08485689166702535778556694051, 6.99239348446402368047701012396, 8.151065434349729714066319725865, 9.115342582428902571028420820495
