L(s) = 1 | + (0.107 + 0.148i)2-s + (1.39 + 0.454i)3-s + (0.607 − 1.87i)4-s + (0.0833 + 0.256i)6-s − 3.26i·7-s + (0.690 − 0.224i)8-s + (−0.674 − 0.489i)9-s + (−1.61 + 1.17i)11-s + (1.70 − 2.34i)12-s + (0.174 − 0.239i)13-s + (0.483 − 0.351i)14-s + (−3.07 − 2.23i)16-s + (4.91 − 1.59i)17-s − 0.152i·18-s + (−0.534 − 1.64i)19-s + ⋯ |
L(s) = 1 | + (0.0761 + 0.104i)2-s + (0.808 + 0.262i)3-s + (0.303 − 0.935i)4-s + (0.0340 + 0.104i)6-s − 1.23i·7-s + (0.244 − 0.0793i)8-s + (−0.224 − 0.163i)9-s + (−0.487 + 0.354i)11-s + (0.491 − 0.675i)12-s + (0.0483 − 0.0665i)13-s + (0.129 − 0.0938i)14-s + (−0.768 − 0.558i)16-s + (1.19 − 0.386i)17-s − 0.0359i·18-s + (−0.122 − 0.377i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.68823 - 1.07138i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.68823 - 1.07138i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.107 - 0.148i)T + (-0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (-1.39 - 0.454i)T + (2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + 3.26iT - 7T^{2} \) |
| 11 | \( 1 + (1.61 - 1.17i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.174 + 0.239i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-4.91 + 1.59i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (0.534 + 1.64i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.516 - 0.711i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.82 - 5.62i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.88 - 5.80i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-4.75 + 6.54i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-0.821 - 0.596i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 3.24iT - 43T^{2} \) |
| 47 | \( 1 + (-4.01 - 1.30i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-7.70 - 2.50i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-4.80 - 3.48i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (0.740 - 0.538i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (6.55 - 2.12i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.84 - 5.67i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-5.19 - 7.14i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.39 + 7.38i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-13.8 + 4.48i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-6.08 + 4.42i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (6.39 + 2.07i)T + (78.4 + 57.0i)T^{2} \) |
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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
−10.36409151496140556157880697354, −9.689799427981636068923929321799, −8.848299450545260192226962759996, −7.63658076862006260315629643141, −7.07561679854809404218738334262, −5.86721655554920213900979372972, −4.88246670727134565260815334959, −3.75430245919133842705565509700, −2.62146289288296814702160782195, −1.01558198900896374371762983638,
2.16079881493560758230595424895, 2.83743101365522621841638051095, 3.83465381180779367684435392312, 5.35970238974976517637603023800, 6.24644644371205914754396279104, 7.65982606490884409063817667610, 8.101359688835410063824223259207, 8.756463813953429874801164297775, 9.713420455715311790434951059948, 10.92014093642117017868341611292