| L(s) = 1 | − 2-s − 4-s + 2·5-s + 7-s + 3·8-s − 2·10-s + 11-s − 6·13-s − 14-s − 16-s − 2·17-s − 2·20-s − 22-s − 25-s + 6·26-s − 28-s − 2·29-s − 8·31-s − 5·32-s + 2·34-s + 2·35-s − 6·37-s + 6·40-s + 10·41-s − 4·43-s − 44-s + 8·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.377·7-s + 1.06·8-s − 0.632·10-s + 0.301·11-s − 1.66·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s − 0.447·20-s − 0.213·22-s − 1/5·25-s + 1.17·26-s − 0.188·28-s − 0.371·29-s − 1.43·31-s − 0.883·32-s + 0.342·34-s + 0.338·35-s − 0.986·37-s + 0.948·40-s + 1.56·41-s − 0.609·43-s − 0.150·44-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.04820991121984, −12.63037060661755, −12.27652087628740, −11.60992026782373, −11.10485746781808, −10.64854131827015, −10.09899379937600, −9.855545298439218, −9.315347804574863, −8.972908115431009, −8.661926371103554, −7.824444260179570, −7.545882672839475, −7.100249711260111, −6.585420250506793, −5.758181632010456, −5.421100092825427, −5.018801362904949, −4.326437739591212, −4.025817970198569, −3.214807752632724, −2.353986043949581, −2.052604558848981, −1.504294291789919, −0.6759407519195875, 0,
0.6759407519195875, 1.504294291789919, 2.052604558848981, 2.353986043949581, 3.214807752632724, 4.025817970198569, 4.326437739591212, 5.018801362904949, 5.421100092825427, 5.758181632010456, 6.585420250506793, 7.100249711260111, 7.545882672839475, 7.824444260179570, 8.661926371103554, 8.972908115431009, 9.315347804574863, 9.855545298439218, 10.09899379937600, 10.64854131827015, 11.10485746781808, 11.60992026782373, 12.27652087628740, 12.63037060661755, 13.04820991121984
