| L(s) = 1 | + 25·5-s + 242·7-s − 656·11-s − 206·13-s − 1.69e3·17-s − 1.36e3·19-s − 2.19e3·23-s + 625·25-s + 2.21e3·29-s − 1.70e3·31-s + 6.05e3·35-s − 846·37-s + 1.81e3·41-s + 1.05e4·43-s − 1.20e4·47-s + 4.17e4·49-s − 3.25e4·53-s − 1.64e4·55-s − 8.66e3·59-s − 3.46e4·61-s − 5.15e3·65-s − 4.75e4·67-s − 948·71-s − 6.31e4·73-s − 1.58e5·77-s + 4.65e4·79-s + 8.87e4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.86·7-s − 1.63·11-s − 0.338·13-s − 1.41·17-s − 0.866·19-s − 0.866·23-s + 1/5·25-s + 0.489·29-s − 0.317·31-s + 0.834·35-s − 0.101·37-s + 0.168·41-s + 0.868·43-s − 0.797·47-s + 2.48·49-s − 1.59·53-s − 0.731·55-s − 0.324·59-s − 1.19·61-s − 0.151·65-s − 1.29·67-s − 0.0223·71-s − 1.38·73-s − 3.05·77-s + 0.838·79-s + 1.41·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 242 T + p^{5} T^{2} \) |
| 11 | \( 1 + 656 T + p^{5} T^{2} \) |
| 13 | \( 1 + 206 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1690 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1364 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2198 T + p^{5} T^{2} \) |
| 29 | \( 1 - 2218 T + p^{5} T^{2} \) |
| 31 | \( 1 + 1700 T + p^{5} T^{2} \) |
| 37 | \( 1 + 846 T + p^{5} T^{2} \) |
| 41 | \( 1 - 1818 T + p^{5} T^{2} \) |
| 43 | \( 1 - 10534 T + p^{5} T^{2} \) |
| 47 | \( 1 + 12074 T + p^{5} T^{2} \) |
| 53 | \( 1 + 32586 T + p^{5} T^{2} \) |
| 59 | \( 1 + 8668 T + p^{5} T^{2} \) |
| 61 | \( 1 + 34670 T + p^{5} T^{2} \) |
| 67 | \( 1 + 47566 T + p^{5} T^{2} \) |
| 71 | \( 1 + 948 T + p^{5} T^{2} \) |
| 73 | \( 1 + 63102 T + p^{5} T^{2} \) |
| 79 | \( 1 - 46536 T + p^{5} T^{2} \) |
| 83 | \( 1 - 88778 T + p^{5} T^{2} \) |
| 89 | \( 1 - 104934 T + p^{5} T^{2} \) |
| 97 | \( 1 + 36254 T + p^{5} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.53214343183816630390645188809, −9.101184860158413775907322248662, −8.169572457160418471297405711331, −7.58265276891097727582643051798, −6.17492577695426818694092441196, −5.03511774749138482163311462575, −4.47074837957027797019045289451, −2.50895899772586183616065402595, −1.75082144137225965024189075519, 0,
1.75082144137225965024189075519, 2.50895899772586183616065402595, 4.47074837957027797019045289451, 5.03511774749138482163311462575, 6.17492577695426818694092441196, 7.58265276891097727582643051798, 8.169572457160418471297405711331, 9.101184860158413775907322248662, 10.53214343183816630390645188809
