| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s + 7-s + 8-s + 9-s − 2·12-s + 13-s + 14-s + 16-s + 18-s + 19-s − 2·21-s + 23-s − 2·24-s + 26-s + 4·27-s + 28-s + 3·29-s + 2·31-s + 32-s + 36-s + 2·37-s + 38-s − 2·39-s − 3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.577·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.235·18-s + 0.229·19-s − 0.436·21-s + 0.208·23-s − 0.408·24-s + 0.196·26-s + 0.769·27-s + 0.188·28-s + 0.557·29-s + 0.359·31-s + 0.176·32-s + 1/6·36-s + 0.328·37-s + 0.162·38-s − 0.320·39-s − 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.79560017885103, −12.99198469716767, −12.67342990407123, −12.19260289679981, −11.75438775128992, −11.23343674345517, −11.05892393239411, −10.49792661920163, −9.965837834164823, −9.438836159494235, −8.739623725710005, −8.171661005836425, −7.747819404936069, −7.052732977076636, −6.511164362006194, −6.226486398741989, −5.625761850573803, −5.166703127982408, −4.658478748970701, −4.336070507313745, −3.417086001362114, −3.033950257715468, −2.235912545940488, −1.481261055154381, −0.8925073952000481, 0,
0.8925073952000481, 1.481261055154381, 2.235912545940488, 3.033950257715468, 3.417086001362114, 4.336070507313745, 4.658478748970701, 5.166703127982408, 5.625761850573803, 6.226486398741989, 6.511164362006194, 7.052732977076636, 7.747819404936069, 8.171661005836425, 8.739623725710005, 9.438836159494235, 9.965837834164823, 10.49792661920163, 11.05892393239411, 11.23343674345517, 11.75438775128992, 12.19260289679981, 12.67342990407123, 12.99198469716767, 13.79560017885103
