| L(s) = 1 | + 2-s + 4-s − 2·5-s + 2·7-s + 8-s − 2·10-s − 4·13-s + 2·14-s + 16-s + 17-s − 2·19-s − 2·20-s − 2·23-s − 25-s − 4·26-s + 2·28-s + 2·29-s − 4·31-s + 32-s + 34-s − 4·35-s + 6·37-s − 2·38-s − 2·40-s + 6·41-s − 2·43-s − 2·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.755·7-s + 0.353·8-s − 0.632·10-s − 1.10·13-s + 0.534·14-s + 1/4·16-s + 0.242·17-s − 0.458·19-s − 0.447·20-s − 0.417·23-s − 1/5·25-s − 0.784·26-s + 0.377·28-s + 0.371·29-s − 0.718·31-s + 0.176·32-s + 0.171·34-s − 0.676·35-s + 0.986·37-s − 0.324·38-s − 0.316·40-s + 0.937·41-s − 0.304·43-s − 0.294·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−15.04125935776284, −14.66901717917753, −14.21949405847779, −13.64826177665495, −12.93089941197154, −12.53426716350738, −11.94304072401693, −11.61795633218787, −11.10641734455305, −10.55488917062247, −9.888092607394366, −9.381274016627211, −8.393495188191614, −8.183902084867787, −7.379158603684280, −7.231860813121985, −6.367333487970951, −5.649421467776598, −5.159727433202811, −4.412626117886973, −4.141549721711078, −3.412680355588220, −2.555018182215442, −2.063194397367315, −1.044157859201548, 0,
1.044157859201548, 2.063194397367315, 2.555018182215442, 3.412680355588220, 4.141549721711078, 4.412626117886973, 5.159727433202811, 5.649421467776598, 6.367333487970951, 7.231860813121985, 7.379158603684280, 8.183902084867787, 8.393495188191614, 9.381274016627211, 9.888092607394366, 10.55488917062247, 11.10641734455305, 11.61795633218787, 11.94304072401693, 12.53426716350738, 12.93089941197154, 13.64826177665495, 14.21949405847779, 14.66901717917753, 15.04125935776284
