| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s − 4·11-s + 12-s − 15-s + 16-s − 6·17-s − 18-s − 4·19-s − 20-s + 4·22-s − 23-s − 24-s + 25-s + 27-s − 2·29-s + 30-s − 32-s − 4·33-s + 6·34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.852·22-s − 0.208·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.182·30-s − 0.176·32-s − 0.696·33-s + 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 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
−14.04807729055553, −13.52863272760490, −13.01563811827790, −12.80507292923062, −12.10602233243385, −11.50604853326089, −11.07157133125891, −10.67583632882276, −10.07095188152199, −9.792408422958696, −8.974162935722501, −8.672695036196743, −8.179570805758699, −7.876930051158846, −7.160774863634649, −6.785337531078083, −6.273363259038292, −5.498757838304471, −4.893319686666648, −4.352940898325920, −3.723409810974249, −3.078920097029389, −2.425087358259283, −2.060865220293744, −1.274467227559031, 0, 0,
1.274467227559031, 2.060865220293744, 2.425087358259283, 3.078920097029389, 3.723409810974249, 4.352940898325920, 4.893319686666648, 5.498757838304471, 6.273363259038292, 6.785337531078083, 7.160774863634649, 7.876930051158846, 8.179570805758699, 8.672695036196743, 8.974162935722501, 9.792408422958696, 10.07095188152199, 10.67583632882276, 11.07157133125891, 11.50604853326089, 12.10602233243385, 12.80507292923062, 13.01563811827790, 13.52863272760490, 14.04807729055553
