L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 2·7-s + 8-s + 9-s + 10-s − 2·11-s + 12-s − 2·13-s − 2·14-s + 15-s + 16-s + 18-s + 19-s + 20-s − 2·21-s − 2·22-s + 6·23-s + 24-s + 25-s − 2·26-s + 27-s − 2·28-s − 29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s − 0.554·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.436·21-s − 0.426·22-s + 1.25·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.377·28-s − 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.681675480\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.681675480\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−13.46699804926879, −12.80378524850850, −12.37910034488476, −12.15350259433143, −11.36166296715059, −10.78576703744431, −10.47492510625433, −9.886797187397627, −9.379146022548405, −9.150345743706595, −8.343524230675362, −7.897496800651768, −7.297167290369019, −6.964161854267438, −6.263228492995590, −5.982061020192026, −5.224591979437724, −4.810711902280315, −4.305659217796979, −3.589704565791828, −2.977837807267449, −2.688229280534689, −2.178286277642364, −1.278546371691470, −0.5963117814094264,
0.5963117814094264, 1.278546371691470, 2.178286277642364, 2.688229280534689, 2.977837807267449, 3.589704565791828, 4.305659217796979, 4.810711902280315, 5.224591979437724, 5.982061020192026, 6.263228492995590, 6.964161854267438, 7.297167290369019, 7.897496800651768, 8.343524230675362, 9.150345743706595, 9.379146022548405, 9.886797187397627, 10.47492510625433, 10.78576703744431, 11.36166296715059, 12.15350259433143, 12.37910034488476, 12.80378524850850, 13.46699804926879