| L(s) = 1 | + 1.73·2-s − 3-s + 0.999·4-s − 5-s − 1.73·6-s − 4.73·7-s − 1.73·8-s + 9-s − 1.73·10-s − 1.46·11-s − 0.999·12-s − 1.26·13-s − 8.19·14-s + 15-s − 5·16-s − 1.46·17-s + 1.73·18-s + 5.46·19-s − 0.999·20-s + 4.73·21-s − 2.53·22-s − 0.535·23-s + 1.73·24-s + 25-s − 2.19·26-s − 27-s − 4.73·28-s + ⋯ |
| L(s) = 1 | + 1.22·2-s − 0.577·3-s + 0.499·4-s − 0.447·5-s − 0.707·6-s − 1.78·7-s − 0.612·8-s + 0.333·9-s − 0.547·10-s − 0.441·11-s − 0.288·12-s − 0.351·13-s − 2.19·14-s + 0.258·15-s − 1.25·16-s − 0.355·17-s + 0.408·18-s + 1.25·19-s − 0.223·20-s + 1.03·21-s − 0.540·22-s − 0.111·23-s + 0.353·24-s + 0.200·25-s − 0.430·26-s − 0.192·27-s − 0.894·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 7 | \( 1 + 4.73T + 7T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 17 | \( 1 + 1.46T + 17T^{2} \) |
| 19 | \( 1 - 5.46T + 19T^{2} \) |
| 23 | \( 1 + 0.535T + 23T^{2} \) |
| 29 | \( 1 - 0.732T + 29T^{2} \) |
| 37 | \( 1 + 6.73T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 3.80T + 71T^{2} \) |
| 73 | \( 1 - 5.66T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 - 7.26T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−10.75112441983124073734164359425, −9.794056020165385720727289982897, −8.998013183599136711827778859149, −7.41818371466859298564908741552, −6.57082876710920788046391442292, −5.75870251084481670459000331645, −4.82891273618117643078569069084, −3.67203822369808089842883042000, −2.89576090844624388716244767869, 0,
2.89576090844624388716244767869, 3.67203822369808089842883042000, 4.82891273618117643078569069084, 5.75870251084481670459000331645, 6.57082876710920788046391442292, 7.41818371466859298564908741552, 8.998013183599136711827778859149, 9.794056020165385720727289982897, 10.75112441983124073734164359425
