| L(s) = 1 | − 0.694·2-s + 9·3-s − 31.5·4-s − 6.25·6-s − 83.1·7-s + 44.1·8-s + 81·9-s + 121·11-s − 283.·12-s + 674.·13-s + 57.7·14-s + 977.·16-s − 1.92e3·17-s − 56.2·18-s − 149.·19-s − 748.·21-s − 84.0·22-s + 1.35e3·23-s + 397.·24-s − 468.·26-s + 729·27-s + 2.62e3·28-s − 7.32e3·29-s − 4.21e3·31-s − 2.09e3·32-s + 1.08e3·33-s + 1.33e3·34-s + ⋯ |
| L(s) = 1 | − 0.122·2-s + 0.577·3-s − 0.984·4-s − 0.0709·6-s − 0.641·7-s + 0.243·8-s + 0.333·9-s + 0.301·11-s − 0.568·12-s + 1.10·13-s + 0.0787·14-s + 0.954·16-s − 1.61·17-s − 0.0409·18-s − 0.0951·19-s − 0.370·21-s − 0.0370·22-s + 0.534·23-s + 0.140·24-s − 0.135·26-s + 0.192·27-s + 0.631·28-s − 1.61·29-s − 0.787·31-s − 0.361·32-s + 0.174·33-s + 0.198·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 + 0.694T + 32T^{2} \) |
| 7 | \( 1 + 83.1T + 1.68e4T^{2} \) |
| 13 | \( 1 - 674.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.92e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 149.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.35e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.21e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.34e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.86e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.20e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.45e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.33e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.58e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.89e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.65e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.10e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.73e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.16e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.52e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.09e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.37e4T + 8.58e9T^{2} \) |
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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
−9.244751937360883475411149572216, −8.428959922781021647355991791509, −7.47924671998887287775658630026, −6.46062820198606362844246076825, −5.52730126559955464673100110921, −4.20850074338068593651839012484, −3.77849809297233710936134131110, −2.49975657489757273391537176730, −1.16379482382485122669916559587, 0,
1.16379482382485122669916559587, 2.49975657489757273391537176730, 3.77849809297233710936134131110, 4.20850074338068593651839012484, 5.52730126559955464673100110921, 6.46062820198606362844246076825, 7.47924671998887287775658630026, 8.428959922781021647355991791509, 9.244751937360883475411149572216
