| L(s) = 1 | − 2-s − 3-s − 4-s + 5-s + 6-s + 3·8-s − 2·9-s − 10-s + 2·11-s + 12-s + 4·13-s − 15-s − 16-s − 3·17-s + 2·18-s − 19-s − 20-s − 2·22-s − 3·24-s − 4·25-s − 4·26-s + 5·27-s − 2·29-s + 30-s − 31-s − 5·32-s − 2·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.06·8-s − 2/3·9-s − 0.316·10-s + 0.603·11-s + 0.288·12-s + 1.10·13-s − 0.258·15-s − 1/4·16-s − 0.727·17-s + 0.471·18-s − 0.229·19-s − 0.223·20-s − 0.426·22-s − 0.612·24-s − 4/5·25-s − 0.784·26-s + 0.962·27-s − 0.371·29-s + 0.182·30-s − 0.179·31-s − 0.883·32-s − 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10051 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10051 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8335168913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8335168913\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−16.81450015996586, −16.25637622514311, −15.73714987599726, −14.85698194949896, −14.23297528847918, −13.77307492802280, −13.20758078727635, −12.66873759194467, −11.79277996350621, −11.21365134803966, −10.81380578810129, −10.17861678259549, −9.287932890897402, −9.141992409848200, −8.360802109330904, −7.872553743490008, −6.922774238915672, −6.210957730065602, −5.772799270358894, −4.971752445762880, −4.196450832308097, −3.553020569656460, −2.343877940747551, −1.439893144691752, −0.5346466362564858,
0.5346466362564858, 1.439893144691752, 2.343877940747551, 3.553020569656460, 4.196450832308097, 4.971752445762880, 5.772799270358894, 6.210957730065602, 6.922774238915672, 7.872553743490008, 8.360802109330904, 9.141992409848200, 9.287932890897402, 10.17861678259549, 10.81380578810129, 11.21365134803966, 11.79277996350621, 12.66873759194467, 13.20758078727635, 13.77307492802280, 14.23297528847918, 14.85698194949896, 15.73714987599726, 16.25637622514311, 16.81450015996586
