| L(s) = 1 | − 2.57·2-s − 0.188·3-s + 4.63·4-s + 4.12·5-s + 0.485·6-s − 2.15·7-s − 6.79·8-s − 2.96·9-s − 10.6·10-s − 11-s − 0.873·12-s + 5.56·14-s − 0.776·15-s + 8.23·16-s − 3.09·17-s + 7.63·18-s + 2.68·19-s + 19.1·20-s + 0.406·21-s + 2.57·22-s + 3.31·23-s + 1.27·24-s + 11.9·25-s + 1.12·27-s − 10.0·28-s − 3.34·29-s + 1.99·30-s + ⋯ |
| L(s) = 1 | − 1.82·2-s − 0.108·3-s + 2.31·4-s + 1.84·5-s + 0.198·6-s − 0.815·7-s − 2.40·8-s − 0.988·9-s − 3.35·10-s − 0.301·11-s − 0.252·12-s + 1.48·14-s − 0.200·15-s + 2.05·16-s − 0.750·17-s + 1.80·18-s + 0.616·19-s + 4.27·20-s + 0.0886·21-s + 0.549·22-s + 0.690·23-s + 0.261·24-s + 2.39·25-s + 0.216·27-s − 1.89·28-s − 0.621·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.57T + 2T^{2} \) |
| 3 | \( 1 + 0.188T + 3T^{2} \) |
| 5 | \( 1 - 4.12T + 5T^{2} \) |
| 7 | \( 1 + 2.15T + 7T^{2} \) |
| 17 | \( 1 + 3.09T + 17T^{2} \) |
| 19 | \( 1 - 2.68T + 19T^{2} \) |
| 23 | \( 1 - 3.31T + 23T^{2} \) |
| 29 | \( 1 + 3.34T + 29T^{2} \) |
| 31 | \( 1 + 9.96T + 31T^{2} \) |
| 37 | \( 1 + 2.05T + 37T^{2} \) |
| 41 | \( 1 + 9.49T + 41T^{2} \) |
| 43 | \( 1 - 2.84T + 43T^{2} \) |
| 47 | \( 1 - 9.46T + 47T^{2} \) |
| 53 | \( 1 + 5.34T + 53T^{2} \) |
| 59 | \( 1 + 4.62T + 59T^{2} \) |
| 61 | \( 1 + 0.193T + 61T^{2} \) |
| 67 | \( 1 - 5.78T + 67T^{2} \) |
| 71 | \( 1 - 3.78T + 71T^{2} \) |
| 73 | \( 1 + 7.47T + 73T^{2} \) |
| 79 | \( 1 + 17.2T + 79T^{2} \) |
| 83 | \( 1 - 0.218T + 83T^{2} \) |
| 89 | \( 1 - 1.52T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−9.168145434971892843487891246002, −8.436924782964661521254339147524, −7.24844229492673919535899447773, −6.66309589205076466061350394509, −5.87929537443495361204080124917, −5.31386429311046780395090554036, −3.14403121527014061723083128372, −2.39975646240554396366547322008, −1.49962905481954392329831094761, 0,
1.49962905481954392329831094761, 2.39975646240554396366547322008, 3.14403121527014061723083128372, 5.31386429311046780395090554036, 5.87929537443495361204080124917, 6.66309589205076466061350394509, 7.24844229492673919535899447773, 8.436924782964661521254339147524, 9.168145434971892843487891246002
