| L(s) = 1 | + 25·5-s + 62·7-s − 144·11-s − 654·13-s + 1.19e3·17-s − 556·19-s + 2.18e3·23-s + 625·25-s + 1.57e3·29-s − 9.66e3·31-s + 1.55e3·35-s − 3.53e3·37-s − 7.46e3·41-s + 7.11e3·43-s − 2.82e4·47-s − 1.29e4·49-s + 1.30e4·53-s − 3.60e3·55-s − 3.70e4·59-s + 3.95e4·61-s − 1.63e4·65-s + 5.67e4·67-s + 4.55e4·71-s + 1.18e4·73-s − 8.92e3·77-s − 9.42e4·79-s − 3.14e4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.478·7-s − 0.358·11-s − 1.07·13-s + 0.998·17-s − 0.353·19-s + 0.860·23-s + 1/5·25-s + 0.348·29-s − 1.80·31-s + 0.213·35-s − 0.424·37-s − 0.693·41-s + 0.586·43-s − 1.86·47-s − 0.771·49-s + 0.637·53-s − 0.160·55-s − 1.38·59-s + 1.36·61-s − 0.479·65-s + 1.54·67-s + 1.07·71-s + 0.260·73-s − 0.171·77-s − 1.69·79-s − 0.501·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - p^{2} T \) |
| good | 7 | \( 1 - 62 T + p^{5} T^{2} \) |
| 11 | \( 1 + 144 T + p^{5} T^{2} \) |
| 13 | \( 1 + 654 T + p^{5} T^{2} \) |
| 17 | \( 1 - 70 p T + p^{5} T^{2} \) |
| 19 | \( 1 + 556 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2182 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1578 T + p^{5} T^{2} \) |
| 31 | \( 1 + 9660 T + p^{5} T^{2} \) |
| 37 | \( 1 + 3534 T + p^{5} T^{2} \) |
| 41 | \( 1 + 182 p T + p^{5} T^{2} \) |
| 43 | \( 1 - 7114 T + p^{5} T^{2} \) |
| 47 | \( 1 + 602 p T + p^{5} T^{2} \) |
| 53 | \( 1 - 13046 T + p^{5} T^{2} \) |
| 59 | \( 1 + 37092 T + p^{5} T^{2} \) |
| 61 | \( 1 - 39570 T + p^{5} T^{2} \) |
| 67 | \( 1 - 56734 T + p^{5} T^{2} \) |
| 71 | \( 1 - 45588 T + p^{5} T^{2} \) |
| 73 | \( 1 - 11842 T + p^{5} T^{2} \) |
| 79 | \( 1 + 94216 T + p^{5} T^{2} \) |
| 83 | \( 1 + 31482 T + p^{5} T^{2} \) |
| 89 | \( 1 - 94054 T + p^{5} T^{2} \) |
| 97 | \( 1 - 23714 T + p^{5} T^{2} \) |
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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
−9.328986791327008631848892802604, −8.313954453833677848221409267963, −7.49601388964997137894529870613, −6.64796382016052533560014200346, −5.39789776532895498852670875712, −4.93240519394286558300465947088, −3.54419658090432035335980387565, −2.43179012278631434012705249831, −1.39841525415430839900162945907, 0,
1.39841525415430839900162945907, 2.43179012278631434012705249831, 3.54419658090432035335980387565, 4.93240519394286558300465947088, 5.39789776532895498852670875712, 6.64796382016052533560014200346, 7.49601388964997137894529870613, 8.313954453833677848221409267963, 9.328986791327008631848892802604
