| L(s) = 1 | + 7-s + 4·11-s − 2·13-s − 2·17-s − 4·23-s − 2·29-s + 8·31-s − 10·37-s − 2·41-s − 4·43-s + 4·47-s + 49-s + 10·53-s − 4·59-s − 2·61-s − 4·67-s + 6·73-s + 4·77-s + 8·79-s − 4·83-s + 6·89-s − 2·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.834·23-s − 0.371·29-s + 1.43·31-s − 1.64·37-s − 0.312·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s + 1.37·53-s − 0.520·59-s − 0.256·61-s − 0.488·67-s + 0.702·73-s + 0.455·77-s + 0.900·79-s − 0.439·83-s + 0.635·89-s − 0.209·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.76535469074182, −14.21146942055081, −13.73789274565158, −13.47548314919319, −12.56023393391812, −12.01307140826628, −11.91325024946501, −11.23933230459368, −10.59662638783557, −10.13031829427842, −9.575685112960708, −8.995475777062247, −8.532491253296493, −8.003206256255523, −7.307447254412039, −6.781814603892671, −6.345239431911128, −5.632760596898357, −5.019047869045910, −4.389356925253638, −3.901115208933034, −3.226320127144857, −2.358277535429687, −1.784345241908456, −1.022485953188644, 0,
1.022485953188644, 1.784345241908456, 2.358277535429687, 3.226320127144857, 3.901115208933034, 4.389356925253638, 5.019047869045910, 5.632760596898357, 6.345239431911128, 6.781814603892671, 7.307447254412039, 8.003206256255523, 8.532491253296493, 8.995475777062247, 9.575685112960708, 10.13031829427842, 10.59662638783557, 11.23933230459368, 11.91325024946501, 12.01307140826628, 12.56023393391812, 13.47548314919319, 13.73789274565158, 14.21146942055081, 14.76535469074182
