| L(s) = 1 | − 2-s + 0.175·3-s + 4-s − 0.284·5-s − 0.175·6-s + 7-s − 8-s − 2.96·9-s + 0.284·10-s + 4.80·11-s + 0.175·12-s + 0.108·13-s − 14-s − 0.0498·15-s + 16-s + 2.45·17-s + 2.96·18-s − 0.284·20-s + 0.175·21-s − 4.80·22-s − 2.80·23-s − 0.175·24-s − 4.91·25-s − 0.108·26-s − 1.04·27-s + 28-s − 0.648·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.101·3-s + 0.5·4-s − 0.127·5-s − 0.0716·6-s + 0.377·7-s − 0.353·8-s − 0.989·9-s + 0.0898·10-s + 1.44·11-s + 0.0506·12-s + 0.0300·13-s − 0.267·14-s − 0.0128·15-s + 0.250·16-s + 0.596·17-s + 0.699·18-s − 0.0635·20-s + 0.0383·21-s − 1.02·22-s − 0.584·23-s − 0.0358·24-s − 0.983·25-s − 0.0212·26-s − 0.201·27-s + 0.188·28-s − 0.120·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 0.175T + 3T^{2} \) |
| 5 | \( 1 + 0.284T + 5T^{2} \) |
| 11 | \( 1 - 4.80T + 11T^{2} \) |
| 13 | \( 1 - 0.108T + 13T^{2} \) |
| 17 | \( 1 - 2.45T + 17T^{2} \) |
| 23 | \( 1 + 2.80T + 23T^{2} \) |
| 29 | \( 1 + 0.648T + 29T^{2} \) |
| 31 | \( 1 + 8.08T + 31T^{2} \) |
| 37 | \( 1 - 2.39T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 - 3.04T + 43T^{2} \) |
| 47 | \( 1 + 2.73T + 47T^{2} \) |
| 53 | \( 1 + 6.93T + 53T^{2} \) |
| 59 | \( 1 - 6.20T + 59T^{2} \) |
| 61 | \( 1 + 5.42T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 1.90T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 - 5.32T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 - 4.83T + 89T^{2} \) |
| 97 | \( 1 - 7.50T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.959576317334287458529750951491, −7.35613042435029377425940143174, −6.43344120309519649996584900765, −5.89670339184423413738771585170, −5.05295582610987117073104588661, −3.89087079605427936094510607211, −3.34674359738136491495279992395, −2.14806984491410479924923779701, −1.35953987527200304401341289896, 0,
1.35953987527200304401341289896, 2.14806984491410479924923779701, 3.34674359738136491495279992395, 3.89087079605427936094510607211, 5.05295582610987117073104588661, 5.89670339184423413738771585170, 6.43344120309519649996584900765, 7.35613042435029377425940143174, 7.959576317334287458529750951491
