| L(s) = 1 | − 210·5-s − 1.01e3·7-s − 1.09e3·11-s − 1.38e3·13-s − 1.47e4·17-s − 3.99e4·19-s + 6.87e4·23-s − 3.40e4·25-s − 1.02e5·29-s − 2.27e5·31-s + 2.13e5·35-s − 1.60e5·37-s − 1.08e4·41-s − 6.30e5·43-s + 4.72e5·47-s + 2.08e5·49-s − 1.49e6·53-s + 2.29e5·55-s − 2.64e6·59-s − 8.27e5·61-s + 2.90e5·65-s − 1.26e5·67-s − 1.41e6·71-s + 9.80e5·73-s + 1.10e6·77-s + 3.56e6·79-s − 5.67e6·83-s + ⋯ |
| L(s) = 1 | − 0.751·5-s − 1.11·7-s − 0.247·11-s − 0.174·13-s − 0.725·17-s − 1.33·19-s + 1.17·23-s − 0.435·25-s − 0.780·29-s − 1.37·31-s + 0.841·35-s − 0.521·37-s − 0.0245·41-s − 1.20·43-s + 0.664·47-s + 0.253·49-s − 1.37·53-s + 0.185·55-s − 1.67·59-s − 0.466·61-s + 0.131·65-s − 0.0511·67-s − 0.469·71-s + 0.294·73-s + 0.276·77-s + 0.813·79-s − 1.08·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.1854243407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1854243407\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 42 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 1016 T + p^{7} T^{2} \) |
| 11 | \( 1 + 1092 T + p^{7} T^{2} \) |
| 13 | \( 1 + 1382 T + p^{7} T^{2} \) |
| 17 | \( 1 + 14706 T + p^{7} T^{2} \) |
| 19 | \( 1 + 39940 T + p^{7} T^{2} \) |
| 23 | \( 1 - 68712 T + p^{7} T^{2} \) |
| 29 | \( 1 + 102570 T + p^{7} T^{2} \) |
| 31 | \( 1 + 227552 T + p^{7} T^{2} \) |
| 37 | \( 1 + 160526 T + p^{7} T^{2} \) |
| 41 | \( 1 + 10842 T + p^{7} T^{2} \) |
| 43 | \( 1 + 630748 T + p^{7} T^{2} \) |
| 47 | \( 1 - 472656 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1494018 T + p^{7} T^{2} \) |
| 59 | \( 1 + 2640660 T + p^{7} T^{2} \) |
| 61 | \( 1 + 827702 T + p^{7} T^{2} \) |
| 67 | \( 1 + 126004 T + p^{7} T^{2} \) |
| 71 | \( 1 + 1414728 T + p^{7} T^{2} \) |
| 73 | \( 1 - 980282 T + p^{7} T^{2} \) |
| 79 | \( 1 - 3566800 T + p^{7} T^{2} \) |
| 83 | \( 1 + 5672892 T + p^{7} T^{2} \) |
| 89 | \( 1 - 11951190 T + p^{7} T^{2} \) |
| 97 | \( 1 - 8682146 T + p^{7} T^{2} \) |
| 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
−9.463731824908515216961730674060, −8.827526262597001267766960005248, −7.76013825108252094368853609151, −6.91707956420586595273310781223, −6.13418061391787092985430271956, −4.90858923845737235291014411642, −3.87926462403899050773717632668, −3.06955430528177305800110865347, −1.85380152507016844334315234486, −0.17380348242044886484045551726,
0.17380348242044886484045551726, 1.85380152507016844334315234486, 3.06955430528177305800110865347, 3.87926462403899050773717632668, 4.90858923845737235291014411642, 6.13418061391787092985430271956, 6.91707956420586595273310781223, 7.76013825108252094368853609151, 8.827526262597001267766960005248, 9.463731824908515216961730674060
