| L(s) = 1 | − 1.17·2-s + 3-s − 0.616·4-s + 5-s − 1.17·6-s − 4.49·7-s + 3.07·8-s + 9-s − 1.17·10-s − 5.81·11-s − 0.616·12-s + 2.32·13-s + 5.29·14-s + 15-s − 2.38·16-s + 1.46·17-s − 1.17·18-s + 3.05·19-s − 0.616·20-s − 4.49·21-s + 6.83·22-s + 3.07·24-s + 25-s − 2.73·26-s + 27-s + 2.77·28-s + 7.50·29-s + ⋯ |
| L(s) = 1 | − 0.831·2-s + 0.577·3-s − 0.308·4-s + 0.447·5-s − 0.480·6-s − 1.70·7-s + 1.08·8-s + 0.333·9-s − 0.371·10-s − 1.75·11-s − 0.177·12-s + 0.644·13-s + 1.41·14-s + 0.258·15-s − 0.596·16-s + 0.355·17-s − 0.277·18-s + 0.700·19-s − 0.137·20-s − 0.981·21-s + 1.45·22-s + 0.628·24-s + 0.200·25-s − 0.535·26-s + 0.192·27-s + 0.524·28-s + 1.39·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 1.17T + 2T^{2} \) |
| 7 | \( 1 + 4.49T + 7T^{2} \) |
| 11 | \( 1 + 5.81T + 11T^{2} \) |
| 13 | \( 1 - 2.32T + 13T^{2} \) |
| 17 | \( 1 - 1.46T + 17T^{2} \) |
| 19 | \( 1 - 3.05T + 19T^{2} \) |
| 29 | \( 1 - 7.50T + 29T^{2} \) |
| 31 | \( 1 - 0.278T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 + 6.03T + 41T^{2} \) |
| 43 | \( 1 - 2.09T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 9.33T + 53T^{2} \) |
| 59 | \( 1 + 2.63T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 - 6.13T + 67T^{2} \) |
| 71 | \( 1 - 1.67T + 71T^{2} \) |
| 73 | \( 1 + 1.23T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 + 0.840T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−7.69075123763251101705310551467, −6.98363906006973803362769377404, −6.24445746674582682606583042257, −5.41688090338627691605885762114, −4.74856530874660031949623925478, −3.59925559949222839292227270829, −3.07714169440888167348196589655, −2.26605263942012270640460399019, −1.03801743643561482568392540611, 0,
1.03801743643561482568392540611, 2.26605263942012270640460399019, 3.07714169440888167348196589655, 3.59925559949222839292227270829, 4.74856530874660031949623925478, 5.41688090338627691605885762114, 6.24445746674582682606583042257, 6.98363906006973803362769377404, 7.69075123763251101705310551467
