| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s − 12-s + 13-s + 15-s + 16-s − 6·17-s − 18-s − 19-s − 20-s + 24-s + 25-s − 26-s − 27-s − 2·29-s − 30-s − 32-s + 6·34-s + 36-s − 6·37-s + 38-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 − 0.288·12-s + 0.277·13-s + 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.371·29-s − 0.182·30-s − 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.986·37-s + 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.62487682085748, −12.18374805958470, −11.70791572456158, −11.31744281273147, −10.94287214703247, −10.49005094461117, −10.21814740083994, −9.456589250963743, −9.072238020671260, −8.746540407925429, −8.179837716808069, −7.718905690184919, −7.214396213061848, −6.756924708110677, −6.404754349104888, −5.849971861053168, −5.358313425451342, −4.627384057475774, −4.384973592288421, −3.668863419957030, −3.176342592143053, −2.453793151039910, −1.887469410795456, −1.361701295661658, −0.5361298073155130, 0,
0.5361298073155130, 1.361701295661658, 1.887469410795456, 2.453793151039910, 3.176342592143053, 3.668863419957030, 4.384973592288421, 4.627384057475774, 5.358313425451342, 5.849971861053168, 6.404754349104888, 6.756924708110677, 7.214396213061848, 7.718905690184919, 8.179837716808069, 8.746540407925429, 9.072238020671260, 9.456589250963743, 10.21814740083994, 10.49005094461117, 10.94287214703247, 11.31744281273147, 11.70791572456158, 12.18374805958470, 12.62487682085748
