| L(s) = 1 | − 4·2-s + 16·4-s + 51.9·5-s + 125.·7-s − 64·8-s − 207.·10-s − 727.·11-s + 169·13-s − 503.·14-s + 256·16-s − 1.75e3·17-s − 1.91e3·19-s + 830.·20-s + 2.90e3·22-s + 3.54e3·23-s − 428.·25-s − 676·26-s + 2.01e3·28-s − 4.86e3·29-s + 2.31e3·31-s − 1.02e3·32-s + 7.01e3·34-s + 6.53e3·35-s + 855.·37-s + 7.64e3·38-s − 3.32e3·40-s + 1.59e4·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.928·5-s + 0.971·7-s − 0.353·8-s − 0.656·10-s − 1.81·11-s + 0.277·13-s − 0.686·14-s + 0.250·16-s − 1.47·17-s − 1.21·19-s + 0.464·20-s + 1.28·22-s + 1.39·23-s − 0.137·25-s − 0.196·26-s + 0.485·28-s − 1.07·29-s + 0.432·31-s − 0.176·32-s + 1.04·34-s + 0.902·35-s + 0.102·37-s + 0.859·38-s − 0.328·40-s + 1.48·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 4T \) |
| 3 | \( 1 \) |
| 13 | \( 1 - 169T \) |
| good | 5 | \( 1 - 51.9T + 3.12e3T^{2} \) |
| 7 | \( 1 - 125.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 727.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 1.75e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.91e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.54e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.86e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.31e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 855.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.59e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.82e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.54e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.92e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.82e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.70e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.13e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.47e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.14e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.02e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.17e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.44e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.23e4T + 8.58e9T^{2} \) |
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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
−10.83210992893769142493137773987, −9.906001270915681546609300347089, −8.775689139800608709658075136895, −8.065877480422518827358619720284, −6.89444137509146616852745549600, −5.69510265828470869796838823837, −4.68417151196206501694114695562, −2.60120610251849185115840731512, −1.73471197844421914850196786930, 0,
1.73471197844421914850196786930, 2.60120610251849185115840731512, 4.68417151196206501694114695562, 5.69510265828470869796838823837, 6.89444137509146616852745549600, 8.065877480422518827358619720284, 8.775689139800608709658075136895, 9.906001270915681546609300347089, 10.83210992893769142493137773987
