| L(s) = 1 | − 3·3-s + 7-s + 6·9-s − 6·11-s − 13-s − 3·21-s − 23-s − 5·25-s − 9·27-s + 3·29-s + 3·31-s + 18·33-s + 8·37-s + 3·39-s + 9·41-s + 4·43-s − 13·47-s + 49-s − 4·53-s + 4·59-s − 2·61-s + 6·63-s − 4·67-s + 3·69-s + 5·71-s + 3·73-s + 15·75-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.377·7-s + 2·9-s − 1.80·11-s − 0.277·13-s − 0.654·21-s − 0.208·23-s − 25-s − 1.73·27-s + 0.557·29-s + 0.538·31-s + 3.13·33-s + 1.31·37-s + 0.480·39-s + 1.40·41-s + 0.609·43-s − 1.89·47-s + 1/7·49-s − 0.549·53-s + 0.520·59-s − 0.256·61-s + 0.755·63-s − 0.488·67-s + 0.361·69-s + 0.593·71-s + 0.351·73-s + 1.73·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−16.88401554013137, −16.25875707813046, −15.93728650755567, −15.42702495495593, −14.72692185992337, −13.91574728058425, −13.13295186252078, −12.85482972735130, −12.13955392719977, −11.64847400224623, −10.95592563001762, −10.74121826007808, −9.903763767631795, −9.641170180086593, −8.349894553749179, −7.784239348006447, −7.321308608340767, −6.342288672491720, −5.938795323166473, −5.238936988983851, −4.774662584834210, −4.155368022257439, −2.895430012364708, −2.042239885094813, −0.8825491924093755, 0,
0.8825491924093755, 2.042239885094813, 2.895430012364708, 4.155368022257439, 4.774662584834210, 5.238936988983851, 5.938795323166473, 6.342288672491720, 7.321308608340767, 7.784239348006447, 8.349894553749179, 9.641170180086593, 9.903763767631795, 10.74121826007808, 10.95592563001762, 11.64847400224623, 12.13955392719977, 12.85482972735130, 13.13295186252078, 13.91574728058425, 14.72692185992337, 15.42702495495593, 15.93728650755567, 16.25875707813046, 16.88401554013137
