| L(s) = 1 | + (−1.20 − 1.08i)2-s + (0.406 − 0.913i)3-s + (0.169 + 1.60i)4-s + (−0.951 + 0.309i)5-s + (−1.47 + 0.658i)6-s + (−0.5 + 0.866i)7-s + (0.587 − 0.809i)8-s + (−0.669 − 0.743i)9-s + (1.47 + 0.658i)10-s + (1.20 + 1.08i)11-s + (1.53 + 0.500i)12-s + (−0.309 + 0.951i)13-s + (1.53 − 0.499i)14-s + (−0.104 + 0.994i)15-s + (−0.406 − 0.913i)17-s + 1.61i·18-s + ⋯ |
| L(s) = 1 | + (−1.20 − 1.08i)2-s + (0.406 − 0.913i)3-s + (0.169 + 1.60i)4-s + (−0.951 + 0.309i)5-s + (−1.47 + 0.658i)6-s + (−0.5 + 0.866i)7-s + (0.587 − 0.809i)8-s + (−0.669 − 0.743i)9-s + (1.47 + 0.658i)10-s + (1.20 + 1.08i)11-s + (1.53 + 0.500i)12-s + (−0.309 + 0.951i)13-s + (1.53 − 0.499i)14-s + (−0.104 + 0.994i)15-s + (−0.406 − 0.913i)17-s + 1.61i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4846273340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4846273340\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.406 + 0.913i)T \) |
| 5 | \( 1 + (0.951 - 0.309i)T \) |
| 13 | \( 1 + (0.309 - 0.951i)T \) |
| good | 2 | \( 1 + (1.20 + 1.08i)T + (0.104 + 0.994i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-1.20 - 1.08i)T + (0.104 + 0.994i)T^{2} \) |
| 17 | \( 1 + (0.406 + 0.913i)T + (-0.669 + 0.743i)T^{2} \) |
| 19 | \( 1 + (-0.913 + 0.406i)T + (0.669 - 0.743i)T^{2} \) |
| 23 | \( 1 + (-0.743 - 0.669i)T + (0.104 + 0.994i)T^{2} \) |
| 29 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 31 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.978 + 0.207i)T + (0.913 - 0.406i)T^{2} \) |
| 41 | \( 1 + (-0.207 - 0.978i)T + (-0.913 + 0.406i)T^{2} \) |
| 43 | \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.743 - 0.669i)T + (0.104 - 0.994i)T^{2} \) |
| 61 | \( 1 + (0.978 + 0.207i)T + (0.913 + 0.406i)T^{2} \) |
| 67 | \( 1 + (0.0646 - 0.614i)T + (-0.978 - 0.207i)T^{2} \) |
| 71 | \( 1 + (0.978 - 0.207i)T^{2} \) |
| 73 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.743 + 0.669i)T + (0.104 + 0.994i)T^{2} \) |
| 97 | \( 1 + (-0.0646 - 0.614i)T + (-0.978 + 0.207i)T^{2} \) |
| 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
−9.788834406496960241744628080176, −9.094449903310391492049129228089, −8.968899856717625176374250238828, −7.52917578029189169580675006564, −7.28691177758162888329487501445, −6.24717886957619948038785543022, −4.46891056564961789220597746543, −3.17515446073533095833048346122, −2.51370663415033563705692659390, −1.29358138860036969335712234091,
0.76847739084472296387364214975, 3.36003344122042134619202223480, 3.95243065873127905691047727228, 5.26402756143772350549754000219, 6.27705794567346312785233075984, 7.20977700351227510639472187847, 7.993319252070138997934276544916, 8.626219164981535591849168763781, 9.181921421859371707868764622201, 10.11742608075968704116609294289
