| L(s) = 1 | + 2.15·2-s + 2.63·4-s + 1.19·5-s − 1.45·7-s + 1.36·8-s + 2.57·10-s − 11-s − 3.89·13-s − 3.12·14-s − 2.33·16-s + 7.17·17-s − 5.21·19-s + 3.15·20-s − 2.15·22-s + 2.26·23-s − 3.56·25-s − 8.37·26-s − 3.82·28-s − 4.93·29-s + 8.33·31-s − 7.74·32-s + 15.4·34-s − 1.74·35-s − 11.5·37-s − 11.2·38-s + 1.63·40-s + 5.40·41-s + ⋯ |
| L(s) = 1 | + 1.52·2-s + 1.31·4-s + 0.535·5-s − 0.549·7-s + 0.482·8-s + 0.815·10-s − 0.301·11-s − 1.07·13-s − 0.836·14-s − 0.582·16-s + 1.74·17-s − 1.19·19-s + 0.705·20-s − 0.458·22-s + 0.472·23-s − 0.712·25-s − 1.64·26-s − 0.723·28-s − 0.916·29-s + 1.49·31-s − 1.36·32-s + 2.64·34-s − 0.294·35-s − 1.90·37-s − 1.82·38-s + 0.258·40-s + 0.843·41-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 | \( 1 - 2.15T + 2T^{2} \) |
| 5 | \( 1 - 1.19T + 5T^{2} \) |
| 7 | \( 1 + 1.45T + 7T^{2} \) |
| 13 | \( 1 + 3.89T + 13T^{2} \) |
| 17 | \( 1 - 7.17T + 17T^{2} \) |
| 19 | \( 1 + 5.21T + 19T^{2} \) |
| 23 | \( 1 - 2.26T + 23T^{2} \) |
| 29 | \( 1 + 4.93T + 29T^{2} \) |
| 31 | \( 1 - 8.33T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 5.40T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 0.334T + 47T^{2} \) |
| 53 | \( 1 - 8.85T + 53T^{2} \) |
| 59 | \( 1 + 5.77T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 1.00T + 67T^{2} \) |
| 71 | \( 1 - 0.864T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 - 4.41T + 79T^{2} \) |
| 83 | \( 1 - 5.26T + 83T^{2} \) |
| 89 | \( 1 - 7.58T + 89T^{2} \) |
| 97 | \( 1 - 9.65T + 97T^{2} \) |
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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
−7.25763836633888293249303166774, −6.48484485053691617623591857866, −5.98892370612043375615072929925, −5.25779043972202751181082950750, −4.83916651394832267054350199820, −3.89020880032679502118414583519, −3.21622879729354225204618772903, −2.56193819550914862355168343357, −1.69052398838147685849345045838, 0,
1.69052398838147685849345045838, 2.56193819550914862355168343357, 3.21622879729354225204618772903, 3.89020880032679502118414583519, 4.83916651394832267054350199820, 5.25779043972202751181082950750, 5.98892370612043375615072929925, 6.48484485053691617623591857866, 7.25763836633888293249303166774
