| L(s) = 1 | + 2-s + 2.62·3-s + 4-s + 2.62·6-s − 1.83·7-s + 8-s + 3.89·9-s + 4.19·11-s + 2.62·12-s − 0.369·13-s − 1.83·14-s + 16-s − 5.08·17-s + 3.89·18-s + 3.55·19-s − 4.81·21-s + 4.19·22-s + 5.62·23-s + 2.62·24-s − 0.369·26-s + 2.34·27-s − 1.83·28-s + 1.20·29-s + 10.1·31-s + 32-s + 11.0·33-s − 5.08·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.51·3-s + 0.5·4-s + 1.07·6-s − 0.692·7-s + 0.353·8-s + 1.29·9-s + 1.26·11-s + 0.757·12-s − 0.102·13-s − 0.489·14-s + 0.250·16-s − 1.23·17-s + 0.917·18-s + 0.816·19-s − 1.04·21-s + 0.893·22-s + 1.17·23-s + 0.535·24-s − 0.0724·26-s + 0.451·27-s − 0.346·28-s + 0.224·29-s + 1.81·31-s + 0.176·32-s + 1.91·33-s − 0.871·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.602843583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.602843583\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - 2.62T + 3T^{2} \) |
| 7 | \( 1 + 1.83T + 7T^{2} \) |
| 11 | \( 1 - 4.19T + 11T^{2} \) |
| 13 | \( 1 + 0.369T + 13T^{2} \) |
| 17 | \( 1 + 5.08T + 17T^{2} \) |
| 19 | \( 1 - 3.55T + 19T^{2} \) |
| 23 | \( 1 - 5.62T + 23T^{2} \) |
| 29 | \( 1 - 1.20T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 41 | \( 1 + 8.01T + 41T^{2} \) |
| 43 | \( 1 + 2.27T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 9.94T + 53T^{2} \) |
| 59 | \( 1 + 5.34T + 59T^{2} \) |
| 61 | \( 1 + 9.79T + 61T^{2} \) |
| 67 | \( 1 - 1.85T + 67T^{2} \) |
| 71 | \( 1 - 2.86T + 71T^{2} \) |
| 73 | \( 1 + 8.09T + 73T^{2} \) |
| 79 | \( 1 - 6.06T + 79T^{2} \) |
| 83 | \( 1 + 8.93T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 6.05T + 97T^{2} \) |
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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
−9.197722515630436105611758493532, −8.575952082750777824426924662200, −7.64417047746911667206271298516, −6.78098513328263379983282590623, −6.31233989340225536201941060685, −4.88412689989903802429346644503, −4.09363201413085186110958710718, −3.23518045657027919867410381213, −2.67121695803865695587107361899, −1.43679127594039923164431537595,
1.43679127594039923164431537595, 2.67121695803865695587107361899, 3.23518045657027919867410381213, 4.09363201413085186110958710718, 4.88412689989903802429346644503, 6.31233989340225536201941060685, 6.78098513328263379983282590623, 7.64417047746911667206271298516, 8.575952082750777824426924662200, 9.197722515630436105611758493532
