L(s) = 1 | + (0.147 + 0.0394i)2-s + (−1.71 − 0.988i)4-s + (0.671 − 0.671i)5-s + (0.383 + 1.43i)7-s + (−0.428 − 0.428i)8-s + (0.125 − 0.0723i)10-s + (1.27 − 4.77i)11-s + (−0.563 − 3.56i)13-s + 0.225i·14-s + (1.93 + 3.34i)16-s + (3.08 − 5.35i)17-s + (−1.68 + 0.451i)19-s + (−1.81 + 0.485i)20-s + (0.376 − 0.652i)22-s + (−3.74 − 6.49i)23-s + ⋯ |
L(s) = 1 | + (0.104 + 0.0278i)2-s + (−0.855 − 0.494i)4-s + (0.300 − 0.300i)5-s + (0.144 + 0.540i)7-s + (−0.151 − 0.151i)8-s + (0.0396 − 0.0228i)10-s + (0.385 − 1.43i)11-s + (−0.156 − 0.987i)13-s + 0.0603i·14-s + (0.482 + 0.835i)16-s + (0.749 − 1.29i)17-s + (−0.386 + 0.103i)19-s + (−0.405 + 0.108i)20-s + (0.0803 − 0.139i)22-s + (−0.781 − 1.35i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.878338 - 0.721449i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.878338 - 0.721449i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 + (0.563 + 3.56i)T \) |
good | 2 | \( 1 + (-0.147 - 0.0394i)T + (1.73 + i)T^{2} \) |
| 5 | \( 1 + (-0.671 + 0.671i)T - 5iT^{2} \) |
| 7 | \( 1 + (-0.383 - 1.43i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.27 + 4.77i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-3.08 + 5.35i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.68 - 0.451i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (3.74 + 6.49i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.18 - 0.681i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.06 + 2.06i)T + 31iT^{2} \) |
| 37 | \( 1 + (1.85 + 0.497i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-9.12 - 2.44i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-6.31 - 3.64i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.42 - 5.42i)T + 47iT^{2} \) |
| 53 | \( 1 - 8.05iT - 53T^{2} \) |
| 59 | \( 1 + (4.49 - 1.20i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (1.54 - 2.67i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.516 - 1.92i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (0.385 + 1.43i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (7.05 - 7.05i)T - 73iT^{2} \) |
| 79 | \( 1 + 4.39T + 79T^{2} \) |
| 83 | \( 1 + (6.90 - 6.90i)T - 83iT^{2} \) |
| 89 | \( 1 + (-2.33 + 8.70i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-12.5 + 3.35i)T + (84.0 - 48.5i)T^{2} \) |
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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
−11.20837672586588466536625550527, −10.30201638234986989710863946368, −9.273517686582285035407963734032, −8.720846884736681549618852000588, −7.68131752483138749870615901691, −5.92296825187062217536034390199, −5.59660083755541495862845078086, −4.33357131093403495345536256170, −2.91121055692667176973755054037, −0.828392325676112274718163783978,
1.92516540685930323722544721102, 3.80766849935747611306102626558, 4.42082147972383250278610821701, 5.78883582712398389641713794068, 7.07913651975373699630646399566, 7.84353910266402746065500778603, 9.038161002678372508312061999266, 9.794213392634675771061677316100, 10.59452454135947790143431278012, 11.97657128824486910828645159562