| L(s) = 1 | + 2-s − 4-s + 4·7-s − 3·8-s + 4·13-s + 4·14-s − 16-s + 6·17-s + 19-s − 5·25-s + 4·26-s − 4·28-s − 2·29-s + 4·31-s + 5·32-s + 6·34-s − 2·37-s + 38-s + 10·41-s − 4·43-s − 8·47-s + 9·49-s − 5·50-s − 4·52-s + 12·53-s − 12·56-s − 2·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 1.51·7-s − 1.06·8-s + 1.10·13-s + 1.06·14-s − 1/4·16-s + 1.45·17-s + 0.229·19-s − 25-s + 0.784·26-s − 0.755·28-s − 0.371·29-s + 0.718·31-s + 0.883·32-s + 1.02·34-s − 0.328·37-s + 0.162·38-s + 1.56·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s − 0.707·50-s − 0.554·52-s + 1.64·53-s − 1.60·56-s − 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20691 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20691 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.813928131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.813928131\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−15.53060618105080, −14.78203144419642, −14.51042141723963, −14.03220584007963, −13.51558178994402, −13.06520424533363, −12.29829612879061, −11.80542711209357, −11.41916857898962, −10.80079631773105, −10.01086435219661, −9.582716665199400, −8.681065484924217, −8.356645960338766, −7.814974775941224, −7.179963953819243, −6.080132306988609, −5.773735147548701, −5.120723199290567, −4.600502853562242, −3.837548452535332, −3.449828582944661, −2.436189274549425, −1.488203418870444, −0.7886652810255925,
0.7886652810255925, 1.488203418870444, 2.436189274549425, 3.449828582944661, 3.837548452535332, 4.600502853562242, 5.120723199290567, 5.773735147548701, 6.080132306988609, 7.179963953819243, 7.814974775941224, 8.356645960338766, 8.681065484924217, 9.582716665199400, 10.01086435219661, 10.80079631773105, 11.41916857898962, 11.80542711209357, 12.29829612879061, 13.06520424533363, 13.51558178994402, 14.03220584007963, 14.51042141723963, 14.78203144419642, 15.53060618105080
