| L(s) = 1 | − 2-s + 4-s − 0.356·5-s + 4.04·7-s − 8-s + 0.356·10-s + 0.911·11-s − 4.04·14-s + 16-s + 2.09·17-s − 4.98·19-s − 0.356·20-s − 0.911·22-s − 8.49·23-s − 4.87·25-s + 4.04·28-s − 8.51·29-s − 10.7·31-s − 32-s − 2.09·34-s − 1.44·35-s − 0.615·37-s + 4.98·38-s + 0.356·40-s + 7.60·41-s − 6.27·43-s + 0.911·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.159·5-s + 1.53·7-s − 0.353·8-s + 0.112·10-s + 0.274·11-s − 1.08·14-s + 0.250·16-s + 0.508·17-s − 1.14·19-s − 0.0798·20-s − 0.194·22-s − 1.77·23-s − 0.974·25-s + 0.765·28-s − 1.58·29-s − 1.93·31-s − 0.176·32-s − 0.359·34-s − 0.244·35-s − 0.101·37-s + 0.809·38-s + 0.0564·40-s + 1.18·41-s − 0.956·43-s + 0.137·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 0.356T + 5T^{2} \) |
| 7 | \( 1 - 4.04T + 7T^{2} \) |
| 11 | \( 1 - 0.911T + 11T^{2} \) |
| 17 | \( 1 - 2.09T + 17T^{2} \) |
| 19 | \( 1 + 4.98T + 19T^{2} \) |
| 23 | \( 1 + 8.49T + 23T^{2} \) |
| 29 | \( 1 + 8.51T + 29T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 + 0.615T + 37T^{2} \) |
| 41 | \( 1 - 7.60T + 41T^{2} \) |
| 43 | \( 1 + 6.27T + 43T^{2} \) |
| 47 | \( 1 + 1.78T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 6.04T + 59T^{2} \) |
| 61 | \( 1 + 3.10T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 0.533T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 6.49T + 83T^{2} \) |
| 89 | \( 1 - 6.49T + 89T^{2} \) |
| 97 | \( 1 + 1.96T + 97T^{2} \) |
| 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
−8.131684930390957908656536105406, −7.88517396292165238930093406755, −7.11669937392378610449011238461, −6.01565439912088904319511871196, −5.43320811743639688314070191549, −4.31627354366123087736257362815, −3.65713551921664261851533644599, −2.05880920347674677370513617734, −1.67729847054278109645170152862, 0,
1.67729847054278109645170152862, 2.05880920347674677370513617734, 3.65713551921664261851533644599, 4.31627354366123087736257362815, 5.43320811743639688314070191549, 6.01565439912088904319511871196, 7.11669937392378610449011238461, 7.88517396292165238930093406755, 8.131684930390957908656536105406
