| L(s) = 1 | + 0.329·2-s + 3-s − 1.89·4-s + 0.329·6-s + 0.941·7-s − 1.28·8-s + 9-s + 0.669·11-s − 1.89·12-s − 0.137·13-s + 0.310·14-s + 3.35·16-s − 5.62·17-s + 0.329·18-s − 4.88·19-s + 0.941·21-s + 0.220·22-s + 5.06·23-s − 1.28·24-s − 0.0452·26-s + 27-s − 1.78·28-s + 8.64·29-s − 1.30·31-s + 3.67·32-s + 0.669·33-s − 1.85·34-s + ⋯ |
| L(s) = 1 | + 0.233·2-s + 0.577·3-s − 0.945·4-s + 0.134·6-s + 0.355·7-s − 0.453·8-s + 0.333·9-s + 0.201·11-s − 0.545·12-s − 0.0380·13-s + 0.0830·14-s + 0.839·16-s − 1.36·17-s + 0.0777·18-s − 1.12·19-s + 0.205·21-s + 0.0470·22-s + 1.05·23-s − 0.261·24-s − 0.00887·26-s + 0.192·27-s − 0.336·28-s + 1.60·29-s − 0.234·31-s + 0.649·32-s + 0.116·33-s − 0.318·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 107 | \( 1 - T \) |
| good | 2 | \( 1 - 0.329T + 2T^{2} \) |
| 7 | \( 1 - 0.941T + 7T^{2} \) |
| 11 | \( 1 - 0.669T + 11T^{2} \) |
| 13 | \( 1 + 0.137T + 13T^{2} \) |
| 17 | \( 1 + 5.62T + 17T^{2} \) |
| 19 | \( 1 + 4.88T + 19T^{2} \) |
| 23 | \( 1 - 5.06T + 23T^{2} \) |
| 29 | \( 1 - 8.64T + 29T^{2} \) |
| 31 | \( 1 + 1.30T + 31T^{2} \) |
| 37 | \( 1 + 9.24T + 37T^{2} \) |
| 41 | \( 1 - 2.59T + 41T^{2} \) |
| 43 | \( 1 + 6.30T + 43T^{2} \) |
| 47 | \( 1 - 6.53T + 47T^{2} \) |
| 53 | \( 1 - 7.92T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 1.26T + 61T^{2} \) |
| 67 | \( 1 + 3.68T + 67T^{2} \) |
| 71 | \( 1 + 8.28T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + 3.01T + 83T^{2} \) |
| 89 | \( 1 - 5.54T + 89T^{2} \) |
| 97 | \( 1 - 4.81T + 97T^{2} \) |
| 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
−7.56366076025772344161560144462, −6.75663832479036748204241618261, −6.17909600795079951973221065905, −5.08124264675546615504639003167, −4.64223914752474559415106704878, −4.01769883661343014910174347093, −3.18009634493544344074614837451, −2.33734365763100750727260862280, −1.29952099391075701276121514297, 0,
1.29952099391075701276121514297, 2.33734365763100750727260862280, 3.18009634493544344074614837451, 4.01769883661343014910174347093, 4.64223914752474559415106704878, 5.08124264675546615504639003167, 6.17909600795079951973221065905, 6.75663832479036748204241618261, 7.56366076025772344161560144462
