| L(s) = 1 | + 5-s − 2·7-s − 13-s − 6.47·17-s + 2·19-s + 2.47·23-s + 25-s + 6.47·29-s + 4.47·31-s − 2·35-s − 6.94·37-s + 10.9·41-s − 4·47-s − 3·49-s − 8.94·53-s − 12.9·59-s + 6.94·61-s − 65-s − 3.52·67-s − 8.94·71-s − 3.52·73-s − 16.9·79-s − 8·83-s − 6.47·85-s + 2·89-s + 2·91-s + 2·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.277·13-s − 1.56·17-s + 0.458·19-s + 0.515·23-s + 0.200·25-s + 1.20·29-s + 0.803·31-s − 0.338·35-s − 1.14·37-s + 1.70·41-s − 0.583·47-s − 0.428·49-s − 1.22·53-s − 1.68·59-s + 0.889·61-s − 0.124·65-s − 0.430·67-s − 1.06·71-s − 0.412·73-s − 1.90·79-s − 0.878·83-s − 0.702·85-s + 0.211·89-s + 0.209·91-s + 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 - 6.47T + 29T^{2} \) |
| 31 | \( 1 - 4.47T + 31T^{2} \) |
| 37 | \( 1 + 6.94T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 8.94T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 6.94T + 61T^{2} \) |
| 67 | \( 1 + 3.52T + 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 + 3.52T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 4.47T + 97T^{2} \) |
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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
−7.956127691723687514956413987112, −7.04440148617422445770769621014, −6.51678412714163866124325940386, −5.90223294776050535297152234303, −4.88562077384429853631661185544, −4.32301125467405934361872935886, −3.12878505555504367208515628429, −2.56978505348243350544367108724, −1.39581459268134997272586510612, 0,
1.39581459268134997272586510612, 2.56978505348243350544367108724, 3.12878505555504367208515628429, 4.32301125467405934361872935886, 4.88562077384429853631661185544, 5.90223294776050535297152234303, 6.51678412714163866124325940386, 7.04440148617422445770769621014, 7.956127691723687514956413987112
