| L(s) = 1 | + 26.3·5-s + 63.2·7-s − 98.2·11-s − 738.·13-s − 250.·17-s + 1.10e3·19-s − 4.40e3·23-s − 2.43e3·25-s + 7.88e3·29-s − 4.61e3·31-s + 1.66e3·35-s + 1.18e4·37-s − 1.00e4·41-s + 7.03e3·43-s − 1.49e4·47-s − 1.28e4·49-s − 2.24e4·53-s − 2.58e3·55-s − 1.08e4·59-s − 1.18e3·61-s − 1.94e4·65-s + 5.91e4·67-s − 1.43e4·71-s − 5.30e4·73-s − 6.21e3·77-s − 3.73e4·79-s − 1.20e5·83-s + ⋯ |
| L(s) = 1 | + 0.470·5-s + 0.487·7-s − 0.244·11-s − 1.21·13-s − 0.209·17-s + 0.700·19-s − 1.73·23-s − 0.778·25-s + 1.74·29-s − 0.861·31-s + 0.229·35-s + 1.42·37-s − 0.936·41-s + 0.580·43-s − 0.985·47-s − 0.761·49-s − 1.09·53-s − 0.115·55-s − 0.404·59-s − 0.0409·61-s − 0.570·65-s + 1.61·67-s − 0.337·71-s − 1.16·73-s − 0.119·77-s − 0.674·79-s − 1.92·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 26.3T + 3.12e3T^{2} \) |
| 7 | \( 1 - 63.2T + 1.68e4T^{2} \) |
| 11 | \( 1 + 98.2T + 1.61e5T^{2} \) |
| 13 | \( 1 + 738.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 250.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.10e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.40e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.88e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.61e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.18e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.00e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.03e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.49e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.24e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.08e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.18e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.91e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.43e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.30e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.73e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.20e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.78e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.06e5T + 8.58e9T^{2} \) |
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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
−10.07173603264989808726576262842, −9.654834890446066644215057379993, −8.293899219943424710910718656811, −7.55123651571994127480931011585, −6.33901350729370476817138475984, −5.30056133150604752703753672681, −4.32213646864380957059259828185, −2.75846139199883843758015648619, −1.66023429767524299783915574209, 0,
1.66023429767524299783915574209, 2.75846139199883843758015648619, 4.32213646864380957059259828185, 5.30056133150604752703753672681, 6.33901350729370476817138475984, 7.55123651571994127480931011585, 8.293899219943424710910718656811, 9.654834890446066644215057379993, 10.07173603264989808726576262842
