L(s) = 1 | + (−0.654 + 0.755i)2-s + (−0.841 + 0.540i)3-s + (−0.142 − 0.989i)4-s + (0.415 + 0.909i)5-s + (0.142 − 0.989i)6-s + (0.170 − 0.0501i)7-s + (0.841 + 0.540i)8-s + (0.415 − 0.909i)9-s + (−0.959 − 0.281i)10-s + (−0.0281 − 0.0324i)11-s + (0.654 + 0.755i)12-s + (−4.70 − 1.38i)13-s + (−0.0739 + 0.161i)14-s + (−0.841 − 0.540i)15-s + (−0.959 + 0.281i)16-s + (−0.635 + 4.41i)17-s + ⋯ |
L(s) = 1 | + (−0.463 + 0.534i)2-s + (−0.485 + 0.312i)3-s + (−0.0711 − 0.494i)4-s + (0.185 + 0.406i)5-s + (0.0580 − 0.404i)6-s + (0.0645 − 0.0189i)7-s + (0.297 + 0.191i)8-s + (0.138 − 0.303i)9-s + (−0.303 − 0.0890i)10-s + (−0.00848 − 0.00979i)11-s + (0.189 + 0.218i)12-s + (−1.30 − 0.383i)13-s + (−0.0197 + 0.0432i)14-s + (−0.217 − 0.139i)15-s + (−0.239 + 0.0704i)16-s + (−0.154 + 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0140437 - 0.459297i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0140437 - 0.459297i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.654 - 0.755i)T \) |
| 3 | \( 1 + (0.841 - 0.540i)T \) |
| 5 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (-1.68 - 4.49i)T \) |
good | 7 | \( 1 + (-0.170 + 0.0501i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (0.0281 + 0.0324i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (4.70 + 1.38i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.635 - 4.41i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.832 - 5.79i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.802 + 5.58i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (8.62 + 5.54i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (0.965 - 2.11i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-1.37 - 3.00i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (4.26 - 2.74i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 9.19T + 47T^{2} \) |
| 53 | \( 1 + (10.0 - 2.94i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-12.6 - 3.70i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (10.1 + 6.49i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (6.02 - 6.95i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-9.86 + 11.3i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (1.21 + 8.45i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-6.03 - 1.77i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (1.27 - 2.79i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (12.3 - 7.90i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (1.37 + 3.01i)T + (-63.5 + 73.3i)T^{2} \) |
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
−10.74280076940738495376635042951, −9.859049771064503200598981497578, −9.501920021952106424010331136947, −8.074871957609658615488868877773, −7.54547079154785335349307486008, −6.39101643826402103991199458748, −5.71811986792493225657723732566, −4.73867385279332970688645580620, −3.46217963615700684675888184537, −1.80645282787028108185270243876,
0.28885465152642374237095520585, 1.87547301934310689834829557665, 3.03186013473753070840855769832, 4.72252085031010407099505666753, 5.18727817307220382444738407685, 6.85869763452976235414248044492, 7.20986606847590500650987637993, 8.523594831847590901742994621869, 9.250789856829646483914171605726, 9.991119337839530432788038093908