| L(s) = 1 | − 11.2·2-s + 93.8·4-s + 25·5-s − 132.·7-s − 693.·8-s − 280.·10-s − 121·11-s − 916.·13-s + 1.48e3·14-s + 4.77e3·16-s + 607.·17-s + 2.34e3·19-s + 2.34e3·20-s + 1.35e3·22-s − 15.5·23-s + 625·25-s + 1.02e4·26-s − 1.24e4·28-s − 837.·29-s + 2.76e3·31-s − 3.13e4·32-s − 6.81e3·34-s − 3.31e3·35-s + 5.91e3·37-s − 2.62e4·38-s − 1.73e4·40-s + 9.47e3·41-s + ⋯ |
| L(s) = 1 | − 1.98·2-s + 2.93·4-s + 0.447·5-s − 1.02·7-s − 3.83·8-s − 0.886·10-s − 0.301·11-s − 1.50·13-s + 2.02·14-s + 4.66·16-s + 0.509·17-s + 1.48·19-s + 1.31·20-s + 0.597·22-s − 0.00612·23-s + 0.200·25-s + 2.98·26-s − 2.99·28-s − 0.184·29-s + 0.516·31-s − 5.41·32-s − 1.01·34-s − 0.457·35-s + 0.710·37-s − 2.95·38-s − 1.71·40-s + 0.880·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 + 11.2T + 32T^{2} \) |
| 7 | \( 1 + 132.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 916.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 607.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.34e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 15.5T + 6.43e6T^{2} \) |
| 29 | \( 1 + 837.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.76e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.91e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.47e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 3.47e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.63e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.05e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.94e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.23e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.83e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.07e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.80e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.32e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.39e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.50e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.36e4T + 8.58e9T^{2} \) |
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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
−9.754532383566439831649159171925, −9.132028020091411700572510595935, −7.87502464769362233679068277065, −7.30276097146960240611661744528, −6.40197005237296371429487681744, −5.43389893762486693998107842319, −3.12923507678653172602144367143, −2.40649493957734057876289756586, −1.04508580084436615602565385499, 0,
1.04508580084436615602565385499, 2.40649493957734057876289756586, 3.12923507678653172602144367143, 5.43389893762486693998107842319, 6.40197005237296371429487681744, 7.30276097146960240611661744528, 7.87502464769362233679068277065, 9.132028020091411700572510595935, 9.754532383566439831649159171925
