| L(s) = 1 | + 1.35·2-s − 9·3-s − 30.1·4-s − 30.1·5-s − 12.1·6-s + 2.34·7-s − 84.1·8-s + 81·9-s − 40.8·10-s + 90.6·11-s + 271.·12-s + 356.·13-s + 3.17·14-s + 271.·15-s + 851.·16-s + 1.28e3·17-s + 109.·18-s + 2.08e3·19-s + 910.·20-s − 21.0·21-s + 122.·22-s − 4.86e3·23-s + 757.·24-s − 2.21e3·25-s + 482.·26-s − 729·27-s − 70.7·28-s + ⋯ |
| L(s) = 1 | + 0.239·2-s − 0.577·3-s − 0.942·4-s − 0.539·5-s − 0.138·6-s + 0.0180·7-s − 0.464·8-s + 0.333·9-s − 0.129·10-s + 0.225·11-s + 0.544·12-s + 0.585·13-s + 0.00432·14-s + 0.311·15-s + 0.831·16-s + 1.07·17-s + 0.0797·18-s + 1.32·19-s + 0.508·20-s − 0.0104·21-s + 0.0540·22-s − 1.91·23-s + 0.268·24-s − 0.708·25-s + 0.140·26-s − 0.192·27-s − 0.0170·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 9T \) |
| 103 | \( 1 + 1.06e4T \) |
| good | 2 | \( 1 - 1.35T + 32T^{2} \) |
| 5 | \( 1 + 30.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 2.34T + 1.68e4T^{2} \) |
| 11 | \( 1 - 90.6T + 1.61e5T^{2} \) |
| 13 | \( 1 - 356.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.28e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.08e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.86e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.16e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 657.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.97e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.38e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.07e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.09e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.43e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.67e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.29e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.79e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.73e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.57e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.14e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.36e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.56e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.48e4T + 8.58e9T^{2} \) |
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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
−10.22930698511452153810686014902, −9.638921277028553117164297985937, −8.362728289164242162700741301144, −7.63858023478296422566950895674, −6.17028932536458881071721997980, −5.35021524617546971847875260589, −4.22066991975675478869055157438, −3.37635386883728808992152731806, −1.22219700874269877549174428727, 0,
1.22219700874269877549174428727, 3.37635386883728808992152731806, 4.22066991975675478869055157438, 5.35021524617546971847875260589, 6.17028932536458881071721997980, 7.63858023478296422566950895674, 8.362728289164242162700741301144, 9.638921277028553117164297985937, 10.22930698511452153810686014902
