L(s) = 1 | + (0.154 − 0.476i)2-s + (−2.50 − 1.81i)3-s + (1.41 + 1.02i)4-s + (−1.25 + 0.910i)6-s + 0.0237·7-s + (1.51 − 1.10i)8-s + (2.02 + 6.24i)9-s + (1.10 − 3.40i)11-s + (−1.67 − 5.14i)12-s + (1.16 + 3.59i)13-s + (0.00368 − 0.0113i)14-s + (0.790 + 2.43i)16-s + (2.93 − 2.12i)17-s + 3.29·18-s + (1.96 − 1.42i)19-s + ⋯ |
L(s) = 1 | + (0.109 − 0.336i)2-s + (−1.44 − 1.04i)3-s + (0.707 + 0.514i)4-s + (−0.511 + 0.371i)6-s + 0.00899·7-s + (0.537 − 0.390i)8-s + (0.676 + 2.08i)9-s + (0.333 − 1.02i)11-s + (−0.482 − 1.48i)12-s + (0.323 + 0.995i)13-s + (0.000984 − 0.00302i)14-s + (0.197 + 0.607i)16-s + (0.710 − 0.516i)17-s + 0.775·18-s + (0.451 − 0.327i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.932494 - 0.804907i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.932494 - 0.804907i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + (-0.154 + 0.476i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (2.50 + 1.81i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 - 0.0237T + 7T^{2} \) |
| 11 | \( 1 + (-1.10 + 3.40i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.16 - 3.59i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.93 + 2.12i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.96 + 1.42i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.530 + 1.63i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (3.12 + 2.26i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-4.86 + 3.53i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (0.114 + 0.351i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (2.41 + 7.42i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 0.174T + 43T^{2} \) |
| 47 | \( 1 + (6.31 + 4.59i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-7.25 - 5.27i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (1.37 + 4.23i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.84 + 8.76i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-3.61 + 2.62i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-7.84 - 5.69i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (1.22 - 3.76i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-7.83 - 5.69i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-7.24 + 5.26i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (5.25 - 16.1i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (2.23 + 1.62i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.99633719781409094941092676647, −9.820154646582532306787683733826, −8.396999086443727892177559296631, −7.48414142437442562452373593965, −6.70370888344442810998449873987, −6.12580010252085538289059483035, −5.06875727912272547017711909496, −3.66709910715840177034708949455, −2.17911362410869480069763892830, −0.927052270515350314768999706134,
1.30064129062877623493258137544, 3.38239430558528254099072496237, 4.68300794483714486367996645673, 5.37836137456050981547779391300, 6.10677623791957227726045257295, 6.90708084351588697380436339308, 7.986517045563230250356587222245, 9.562776680807199387913509520668, 10.14190803607685537506829450286, 10.67937186383175854999451891630