| L(s) = 1 | − 2·5-s − 3·7-s − 3·9-s − 2·11-s − 4·13-s − 6·17-s − 7·19-s − 9·23-s − 25-s + 6·29-s − 8·31-s + 6·35-s − 2·37-s − 4·41-s − 5·43-s + 6·45-s − 6·47-s + 2·49-s + 6·53-s + 4·55-s − 14·59-s + 2·61-s + 9·63-s + 8·65-s − 73-s + 6·77-s − 5·79-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.13·7-s − 9-s − 0.603·11-s − 1.10·13-s − 1.45·17-s − 1.60·19-s − 1.87·23-s − 1/5·25-s + 1.11·29-s − 1.43·31-s + 1.01·35-s − 0.328·37-s − 0.624·41-s − 0.762·43-s + 0.894·45-s − 0.875·47-s + 2/7·49-s + 0.824·53-s + 0.539·55-s − 1.82·59-s + 0.256·61-s + 1.13·63-s + 0.992·65-s − 0.117·73-s + 0.683·77-s − 0.562·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 73 | \( 1 + T \) |
| 137 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.90842764591167, −13.47530414767086, −12.99620822197296, −12.48725129111045, −12.13706303941449, −11.70740522957342, −11.14890489962155, −10.66377686789448, −10.19882775477563, −9.772858249961354, −9.099672578762616, −8.696944563706997, −8.141496243000739, −7.871221547571744, −7.143210682741698, −6.544661302057083, −6.372202823661054, −5.648929495344461, −5.077361643162263, −4.406095971544007, −4.018367103277625, −3.425223339107862, −2.758255451771243, −2.340593555966614, −1.747012679694378, 0, 0, 0,
1.747012679694378, 2.340593555966614, 2.758255451771243, 3.425223339107862, 4.018367103277625, 4.406095971544007, 5.077361643162263, 5.648929495344461, 6.372202823661054, 6.544661302057083, 7.143210682741698, 7.871221547571744, 8.141496243000739, 8.696944563706997, 9.099672578762616, 9.772858249961354, 10.19882775477563, 10.66377686789448, 11.14890489962155, 11.70740522957342, 12.13706303941449, 12.48725129111045, 12.99620822197296, 13.47530414767086, 13.90842764591167
