| L(s) = 1 | + 2-s + 4-s − 2.64·5-s + 8-s − 2.64·10-s − 2·11-s + 2.64·13-s + 16-s + 7.93·17-s − 5.29·19-s − 2.64·20-s − 2·22-s − 6·23-s + 2.00·25-s + 2.64·26-s − 5·29-s + 5.29·31-s + 32-s + 7.93·34-s + 3·37-s − 5.29·38-s − 2.64·40-s − 8·43-s − 2·44-s − 6·46-s + 5.29·47-s + 2.00·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.18·5-s + 0.353·8-s − 0.836·10-s − 0.603·11-s + 0.733·13-s + 0.250·16-s + 1.92·17-s − 1.21·19-s − 0.591·20-s − 0.426·22-s − 1.25·23-s + 0.400·25-s + 0.518·26-s − 0.928·29-s + 0.950·31-s + 0.176·32-s + 1.36·34-s + 0.493·37-s − 0.858·38-s − 0.418·40-s − 1.21·43-s − 0.301·44-s − 0.884·46-s + 0.771·47-s + 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2.64T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 2.64T + 13T^{2} \) |
| 17 | \( 1 - 7.93T + 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 5.29T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 5.29T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 5.29T + 59T^{2} \) |
| 61 | \( 1 + 2.64T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−7.59149736753746230244187243491, −6.77697408506253346937432174757, −5.95719090748019658184313200660, −5.45841558728034739520988606801, −4.48987456719487451942679534994, −3.88926603157284258035032168233, −3.37600162599372598172956091518, −2.44933496920508504922044262991, −1.31386878473993661309267904865, 0,
1.31386878473993661309267904865, 2.44933496920508504922044262991, 3.37600162599372598172956091518, 3.88926603157284258035032168233, 4.48987456719487451942679534994, 5.45841558728034739520988606801, 5.95719090748019658184313200660, 6.77697408506253346937432174757, 7.59149736753746230244187243491
