| L(s) = 1 | + 7-s − 2·11-s − 2·13-s + 8·17-s + 2·19-s + 6·29-s − 6·31-s − 8·37-s − 6·41-s + 8·43-s − 4·47-s + 49-s − 2·53-s − 8·59-s + 10·61-s − 12·67-s − 14·71-s + 10·73-s − 2·77-s − 4·79-s − 16·83-s − 10·89-s − 2·91-s − 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.603·11-s − 0.554·13-s + 1.94·17-s + 0.458·19-s + 1.11·29-s − 1.07·31-s − 1.31·37-s − 0.937·41-s + 1.21·43-s − 0.583·47-s + 1/7·49-s − 0.274·53-s − 1.04·59-s + 1.28·61-s − 1.46·67-s − 1.66·71-s + 1.17·73-s − 0.227·77-s − 0.450·79-s − 1.75·83-s − 1.05·89-s − 0.209·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.67738917211530, −15.05369083301398, −14.43030292954165, −14.14899530028902, −13.61656976725821, −12.76327608751835, −12.44063616041881, −11.89389241100467, −11.39063617314521, −10.60617247925142, −10.16800443613483, −9.782897032410292, −8.980997781284033, −8.427343676196891, −7.742038555349214, −7.424308077675515, −6.776855360637956, −5.834112302330627, −5.405889157745384, −4.906960637788748, −4.120041764605754, −3.247406696341786, −2.856599127928945, −1.815921359166709, −1.132973108555356, 0,
1.132973108555356, 1.815921359166709, 2.856599127928945, 3.247406696341786, 4.120041764605754, 4.906960637788748, 5.405889157745384, 5.834112302330627, 6.776855360637956, 7.424308077675515, 7.742038555349214, 8.427343676196891, 8.980997781284033, 9.782897032410292, 10.16800443613483, 10.60617247925142, 11.39063617314521, 11.89389241100467, 12.44063616041881, 12.76327608751835, 13.61656976725821, 14.14899530028902, 14.43030292954165, 15.05369083301398, 15.67738917211530
