| L(s) = 1 | + (−1.39 − 0.208i)2-s + (1.33 + 2.87i)3-s + (1.91 + 0.582i)4-s + (−1.93 − 1.35i)5-s + (−1.27 − 4.29i)6-s + (2.32 + 1.34i)7-s + (−2.55 − 1.21i)8-s + (−4.52 + 5.39i)9-s + (2.42 + 2.29i)10-s + (4.37 + 1.17i)11-s + (0.888 + 6.27i)12-s + (1.64 + 0.765i)13-s + (−2.97 − 2.36i)14-s + (1.29 − 7.36i)15-s + (3.32 + 2.23i)16-s + (−3.06 + 2.56i)17-s + ⋯ |
| L(s) = 1 | + (−0.989 − 0.147i)2-s + (0.773 + 1.65i)3-s + (0.956 + 0.291i)4-s + (−0.864 − 0.605i)5-s + (−0.520 − 1.75i)6-s + (0.880 + 0.508i)7-s + (−0.903 − 0.429i)8-s + (−1.50 + 1.79i)9-s + (0.765 + 0.725i)10-s + (1.31 + 0.353i)11-s + (0.256 + 1.81i)12-s + (0.455 + 0.212i)13-s + (−0.795 − 0.632i)14-s + (0.335 − 1.90i)15-s + (0.830 + 0.557i)16-s + (−0.742 + 0.623i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.589076 + 0.805891i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.589076 + 0.805891i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 + 0.208i)T \) |
| 19 | \( 1 + (3.77 - 2.17i)T \) |
| good | 3 | \( 1 + (-1.33 - 2.87i)T + (-1.92 + 2.29i)T^{2} \) |
| 5 | \( 1 + (1.93 + 1.35i)T + (1.71 + 4.69i)T^{2} \) |
| 7 | \( 1 + (-2.32 - 1.34i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.37 - 1.17i)T + (9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (-1.64 - 0.765i)T + (8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (3.06 - 2.56i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (3.38 + 0.596i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.418 + 4.78i)T + (-28.5 + 5.03i)T^{2} \) |
| 31 | \( 1 + (-0.178 + 0.308i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.07 + 2.07i)T - 37iT^{2} \) |
| 41 | \( 1 + (-0.0566 + 0.155i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-8.09 - 5.66i)T + (14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (-8.16 - 6.85i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.24 - 1.77i)T + (-18.1 + 49.8i)T^{2} \) |
| 59 | \( 1 + (-0.715 + 8.17i)T + (-58.1 - 10.2i)T^{2} \) |
| 61 | \( 1 + (-11.3 + 7.91i)T + (20.8 - 57.3i)T^{2} \) |
| 67 | \( 1 + (0.0145 + 0.166i)T + (-65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (-0.853 + 0.150i)T + (66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (3.00 - 8.24i)T + (-55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (16.0 + 5.84i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-16.6 + 4.46i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-4.38 - 12.0i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (0.0803 - 0.0674i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.51787886855318208069265314551, −11.00640226173566127147183336891, −9.912762352925958902212230964921, −9.043477276899758670180758054019, −8.512553125577300533000795896867, −7.895450232030810993126557469842, −6.08785261236808152762306789385, −4.36107778708802105999620813224, −3.91681793236299583803026551977, −2.12656960273993604806492872967,
0.981680606273108644925195150364, 2.30027614669476244410161999166, 3.73035633735545455227919928496, 6.17798876115304367843736363696, 7.07958591043558981990024661292, 7.48051477299135090791049178079, 8.532459315231614395166203716856, 8.957080269953926067555998135127, 10.72112564113946586135911770078, 11.53125219946346911188867830811
