| L(s) = 1 | − 11·2-s − 9·3-s + 89·4-s + 6·5-s + 99·6-s − 176·7-s − 627·8-s + 81·9-s − 66·10-s − 496·11-s − 801·12-s + 178·13-s + 1.93e3·14-s − 54·15-s + 4.04e3·16-s + 202·17-s − 891·18-s + 534·20-s + 1.58e3·21-s + 5.45e3·22-s + 4.39e3·23-s + 5.64e3·24-s − 3.08e3·25-s − 1.95e3·26-s − 729·27-s − 1.56e4·28-s + 5.90e3·29-s + ⋯ |
| L(s) = 1 | − 1.94·2-s − 0.577·3-s + 2.78·4-s + 0.107·5-s + 1.12·6-s − 1.35·7-s − 3.46·8-s + 1/3·9-s − 0.208·10-s − 1.23·11-s − 1.60·12-s + 0.292·13-s + 2.63·14-s − 0.0619·15-s + 3.95·16-s + 0.169·17-s − 0.648·18-s + 0.298·20-s + 0.783·21-s + 2.40·22-s + 1.73·23-s + 1.99·24-s − 0.988·25-s − 0.568·26-s − 0.192·27-s − 3.77·28-s + 1.30·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1915638284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1915638284\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{2} T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 11 T + p^{5} T^{2} \) |
| 5 | \( 1 - 6 T + p^{5} T^{2} \) |
| 7 | \( 1 + 176 T + p^{5} T^{2} \) |
| 11 | \( 1 + 496 T + p^{5} T^{2} \) |
| 13 | \( 1 - 178 T + p^{5} T^{2} \) |
| 17 | \( 1 - 202 T + p^{5} T^{2} \) |
| 23 | \( 1 - 4396 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5902 T + p^{5} T^{2} \) |
| 31 | \( 1 + 5760 T + p^{5} T^{2} \) |
| 37 | \( 1 - 3906 T + p^{5} T^{2} \) |
| 41 | \( 1 + 15774 T + p^{5} T^{2} \) |
| 43 | \( 1 + 7492 T + p^{5} T^{2} \) |
| 47 | \( 1 + 7452 T + p^{5} T^{2} \) |
| 53 | \( 1 - 29014 T + p^{5} T^{2} \) |
| 59 | \( 1 + 13604 T + p^{5} T^{2} \) |
| 61 | \( 1 + 12466 T + p^{5} T^{2} \) |
| 67 | \( 1 + 43436 T + p^{5} T^{2} \) |
| 71 | \( 1 + 28800 T + p^{5} T^{2} \) |
| 73 | \( 1 - 80746 T + p^{5} T^{2} \) |
| 79 | \( 1 + 76456 T + p^{5} T^{2} \) |
| 83 | \( 1 + 56880 T + p^{5} T^{2} \) |
| 89 | \( 1 - 103266 T + p^{5} T^{2} \) |
| 97 | \( 1 + 82490 T + p^{5} T^{2} \) |
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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
−9.230837949553101819544264054126, −8.479332976153336234821789124284, −7.53777529137282696099431555355, −6.85642139949342265882593230155, −6.18159206849484436597535026360, −5.28557373129483663195806586323, −3.33458507810404999349395751418, −2.58319887526859354826789805826, −1.32778140209485255123253892036, −0.27555143314703734825203014013,
0.27555143314703734825203014013, 1.32778140209485255123253892036, 2.58319887526859354826789805826, 3.33458507810404999349395751418, 5.28557373129483663195806586323, 6.18159206849484436597535026360, 6.85642139949342265882593230155, 7.53777529137282696099431555355, 8.479332976153336234821789124284, 9.230837949553101819544264054126
