| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 4·7-s − 8-s + 9-s + 10-s − 12-s − 2·13-s − 4·14-s + 15-s + 16-s − 6·17-s − 18-s + 4·19-s − 20-s − 4·21-s + 24-s + 25-s + 2·26-s − 27-s + 4·28-s + 6·29-s − 30-s + 8·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.554·13-s − 1.06·14-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 − 0.872·21-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.755·28-s + 1.11·29-s − 0.182·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 41 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−14.98139614175195, −14.32796917770696, −13.71652012534002, −13.36033643531488, −12.32539360860126, −12.09788450036500, −11.57450012928400, −11.17212427469370, −10.76214311062696, −10.17009658044904, −9.629417047601413, −8.953699368183665, −8.360607591417932, −8.044233999634391, −7.459730931170358, −6.855321544899906, −6.455969333388876, −5.585576633390205, −4.995811958718539, −4.546138467950438, −4.044613355052554, −2.907177173959177, −2.383142441438017, −1.511183778078476, −0.9511257119263556, 0,
0.9511257119263556, 1.511183778078476, 2.383142441438017, 2.907177173959177, 4.044613355052554, 4.546138467950438, 4.995811958718539, 5.585576633390205, 6.455969333388876, 6.855321544899906, 7.459730931170358, 8.044233999634391, 8.360607591417932, 8.953699368183665, 9.629417047601413, 10.17009658044904, 10.76214311062696, 11.17212427469370, 11.57450012928400, 12.09788450036500, 12.32539360860126, 13.36033643531488, 13.71652012534002, 14.32796917770696, 14.98139614175195
