L(s) = 1 | + 4.05i·2-s − 5.19·3-s − 0.408·4-s − 16.4·5-s − 21.0i·6-s + 47.3·7-s + 63.1i·8-s + 27·9-s − 66.4i·10-s + 25.6i·11-s + 2.12·12-s + 105. i·13-s + 191. i·14-s + 85.2·15-s − 262.·16-s + 441.·17-s + ⋯ |
L(s) = 1 | + 1.01i·2-s − 0.577·3-s − 0.0255·4-s − 0.656·5-s − 0.584i·6-s + 0.965·7-s + 0.986i·8-s + 0.333·9-s − 0.664i·10-s + 0.211i·11-s + 0.0147·12-s + 0.624i·13-s + 0.977i·14-s + 0.378·15-s − 1.02·16-s + 1.52·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.093068079\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.093068079\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 5.19T \) |
| 59 | \( 1 + (3.44e3 - 496. i)T \) |
good | 2 | \( 1 - 4.05iT - 16T^{2} \) |
| 5 | \( 1 + 16.4T + 625T^{2} \) |
| 7 | \( 1 - 47.3T + 2.40e3T^{2} \) |
| 11 | \( 1 - 25.6iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 105. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 441.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 560.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 764. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.38e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + 950. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 632. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.02e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.95e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 4.32e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.11e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 4.27e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 867. iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 236.T + 2.54e7T^{2} \) |
| 73 | \( 1 + 6.21e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 8.91e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.08e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 6.26e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 7.89e3iT - 8.85e7T^{2} \) |
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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
−12.29985973539582861282746441051, −11.49693088725257956274285029812, −10.85797496854285970601677540063, −9.321824767081502069574722237309, −7.85866170235033092477989596110, −7.57347769336835160841749279773, −6.19299763383188014338367716955, −5.26865439749037792450966131092, −4.04832678505045336065137196356, −1.78538111434115335597177332395,
0.42672401791842696483040857210, 1.83642194599238158442552633239, 3.45664877496578520931764973150, 4.63223307143711017873795913018, 6.04599066410695034494593276879, 7.42903286639907644257245079934, 8.392883205671525682111602006555, 9.951445205183199197191885828715, 10.84508951815289901159108150291, 11.34129634624737799231243997808