| L(s) = 1 | − 27·3-s − 160·5-s − 974·7-s + 729·9-s + 3.95e3·11-s − 574·13-s + 4.32e3·15-s − 8.47e3·17-s − 5.33e4·19-s + 2.62e4·21-s + 9.84e4·23-s − 5.25e4·25-s − 1.96e4·27-s − 5.10e4·29-s − 2.05e5·31-s − 1.06e5·33-s + 1.55e5·35-s − 2.55e5·37-s + 1.54e4·39-s − 6.65e5·41-s − 3.96e5·43-s − 1.16e5·45-s − 5.49e5·47-s + 1.25e5·49-s + 2.28e5·51-s + 7.20e5·53-s − 6.32e5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.572·5-s − 1.07·7-s + 1/3·9-s + 0.896·11-s − 0.0724·13-s + 0.330·15-s − 0.418·17-s − 1.78·19-s + 0.619·21-s + 1.68·23-s − 0.672·25-s − 0.192·27-s − 0.388·29-s − 1.23·31-s − 0.517·33-s + 0.614·35-s − 0.829·37-s + 0.0418·39-s − 1.50·41-s − 0.761·43-s − 0.190·45-s − 0.771·47-s + 0.151·49-s + 0.241·51-s + 0.664·53-s − 0.512·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.5749373019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5749373019\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{3} T \) |
| good | 5 | \( 1 + 32 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 974 T + p^{7} T^{2} \) |
| 11 | \( 1 - 3956 T + p^{7} T^{2} \) |
| 13 | \( 1 + 574 T + p^{7} T^{2} \) |
| 17 | \( 1 + 8474 T + p^{7} T^{2} \) |
| 19 | \( 1 + 53312 T + p^{7} T^{2} \) |
| 23 | \( 1 - 98468 T + p^{7} T^{2} \) |
| 29 | \( 1 + 51060 T + p^{7} T^{2} \) |
| 31 | \( 1 + 205014 T + p^{7} T^{2} \) |
| 37 | \( 1 + 255674 T + p^{7} T^{2} \) |
| 41 | \( 1 + 665394 T + p^{7} T^{2} \) |
| 43 | \( 1 + 396840 T + p^{7} T^{2} \) |
| 47 | \( 1 + 549388 T + p^{7} T^{2} \) |
| 53 | \( 1 - 720060 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1043956 T + p^{7} T^{2} \) |
| 61 | \( 1 - 2055734 T + p^{7} T^{2} \) |
| 67 | \( 1 - 2092652 T + p^{7} T^{2} \) |
| 71 | \( 1 - 2723868 T + p^{7} T^{2} \) |
| 73 | \( 1 + 5190010 T + p^{7} T^{2} \) |
| 79 | \( 1 + 3647110 T + p^{7} T^{2} \) |
| 83 | \( 1 - 2101956 T + p^{7} T^{2} \) |
| 89 | \( 1 - 522514 T + p^{7} T^{2} \) |
| 97 | \( 1 + 1361990 T + p^{7} T^{2} \) |
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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
−10.22986936842466211020131030762, −9.241991274524385398689846262225, −8.435833728593323931556577383277, −6.92382066630651037462300952305, −6.64486625459393967795364132527, −5.38668217008109920087371845282, −4.17291590920838793418024830920, −3.36768486327415163773325310855, −1.82624155700264886896827382677, −0.35307205276417996252778441474,
0.35307205276417996252778441474, 1.82624155700264886896827382677, 3.36768486327415163773325310855, 4.17291590920838793418024830920, 5.38668217008109920087371845282, 6.64486625459393967795364132527, 6.92382066630651037462300952305, 8.435833728593323931556577383277, 9.241991274524385398689846262225, 10.22986936842466211020131030762
