| L(s) = 1 | + 1.25·2-s − 0.433·4-s − 0.278·5-s + 0.0619·7-s − 3.04·8-s − 0.348·10-s − 11-s + 3.85·13-s + 0.0775·14-s − 2.94·16-s + 5.03·17-s − 6.09·19-s + 0.120·20-s − 1.25·22-s + 6.96·23-s − 4.92·25-s + 4.82·26-s − 0.0268·28-s − 6.95·29-s − 1.51·31-s + 2.40·32-s + 6.29·34-s − 0.0172·35-s + 2.79·37-s − 7.62·38-s + 0.848·40-s − 3.82·41-s + ⋯ |
| L(s) = 1 | + 0.884·2-s − 0.216·4-s − 0.124·5-s + 0.0234·7-s − 1.07·8-s − 0.110·10-s − 0.301·11-s + 1.06·13-s + 0.0207·14-s − 0.736·16-s + 1.22·17-s − 1.39·19-s + 0.0270·20-s − 0.266·22-s + 1.45·23-s − 0.984·25-s + 0.945·26-s − 0.00507·28-s − 1.29·29-s − 0.271·31-s + 0.425·32-s + 1.07·34-s − 0.00292·35-s + 0.459·37-s − 1.23·38-s + 0.134·40-s − 0.597·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 1.25T + 2T^{2} \) |
| 5 | \( 1 + 0.278T + 5T^{2} \) |
| 7 | \( 1 - 0.0619T + 7T^{2} \) |
| 13 | \( 1 - 3.85T + 13T^{2} \) |
| 17 | \( 1 - 5.03T + 17T^{2} \) |
| 19 | \( 1 + 6.09T + 19T^{2} \) |
| 23 | \( 1 - 6.96T + 23T^{2} \) |
| 29 | \( 1 + 6.95T + 29T^{2} \) |
| 31 | \( 1 + 1.51T + 31T^{2} \) |
| 37 | \( 1 - 2.79T + 37T^{2} \) |
| 41 | \( 1 + 3.82T + 41T^{2} \) |
| 43 | \( 1 - 1.62T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 - 1.46T + 59T^{2} \) |
| 61 | \( 1 - 1.07T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 + 7.53T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 7.13T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−7.56001853689368834415706226970, −6.48120878476304133768014994408, −6.01507659417225183137905905958, −5.32723969966438875735651292528, −4.69014186530087461712195203786, −3.79451894832312826654236695694, −3.45441855388249658227928321625, −2.46841187294074931626831518035, −1.30710722920723369200914032226, 0,
1.30710722920723369200914032226, 2.46841187294074931626831518035, 3.45441855388249658227928321625, 3.79451894832312826654236695694, 4.69014186530087461712195203786, 5.32723969966438875735651292528, 6.01507659417225183137905905958, 6.48120878476304133768014994408, 7.56001853689368834415706226970
