| L(s) = 1 | + (−3.23 + 2.35i)2-s + (−6.86 + 21.1i)3-s + (4.94 − 15.2i)4-s + (−38.8 − 40.2i)5-s + (−27.4 − 84.4i)6-s − 78.8·7-s + (19.7 + 60.8i)8-s + (−202. − 147. i)9-s + (220. + 38.8i)10-s + (267. − 194. i)11-s + (287. + 208. i)12-s + (374. + 272. i)13-s + (255. − 185. i)14-s + (1.11e3 − 543. i)15-s + (−207. − 150. i)16-s + (−93.8 − 288. i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (−0.440 + 1.35i)3-s + (0.154 − 0.475i)4-s + (−0.694 − 0.719i)5-s + (−0.311 − 0.958i)6-s − 0.608·7-s + (0.109 + 0.336i)8-s + (−0.833 − 0.605i)9-s + (0.696 + 0.122i)10-s + (0.667 − 0.484i)11-s + (0.576 + 0.418i)12-s + (0.614 + 0.446i)13-s + (0.347 − 0.252i)14-s + (1.28 − 0.624i)15-s + (−0.202 − 0.146i)16-s + (−0.0787 − 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.426591 - 0.193654i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.426591 - 0.193654i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.23 - 2.35i)T \) |
| 5 | \( 1 + (38.8 + 40.2i)T \) |
| good | 3 | \( 1 + (6.86 - 21.1i)T + (-196. - 142. i)T^{2} \) |
| 7 | \( 1 + 78.8T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-267. + 194. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-374. - 272. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (93.8 + 288. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (790. + 2.43e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-2.31e3 + 1.67e3i)T + (1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (432. - 1.33e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (2.39e3 + 7.38e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (5.28e3 + 3.83e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (7.08e3 + 5.14e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 1.33e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-9.12e3 + 2.80e4i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (6.15e3 - 1.89e4i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-2.46e4 - 1.78e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-2.36e4 + 1.71e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (1.96e4 + 6.05e4i)T + (-1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (6.17e3 - 1.90e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-7.27e3 + 5.28e3i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (1.68e4 - 5.18e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-1.56e4 - 4.81e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-3.87e4 + 2.81e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (4.45e4 - 1.36e5i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
| 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
−15.02737826088784737176314401380, −13.29826164111251151706017102024, −11.61663447240611338496919039604, −10.77747664043884706617313754344, −9.333910709441697283619027087827, −8.733942270436046622808683000936, −6.76284230200759945917809534167, −5.13974685221657319707218851413, −3.85253835436431669967057405551, −0.32462404081097080345852405172,
1.46393903099949210989349477397, 3.44179501478714700315774000675, 6.31820014044418761282857838617, 7.19951574852092319832482208281, 8.362167686224916744778358526744, 10.14511411456820107875817442066, 11.38323551317775518609048517337, 12.28022655636589987341057571489, 13.11619373699328023779168799526, 14.62862609870329553319378389826
