| L(s) = 1 | + (−1.23 + 0.690i)2-s + (−2.50 − 1.16i)3-s + (1.04 − 1.70i)4-s + (1.02 + 1.47i)5-s + (3.90 − 0.289i)6-s + (−0.843 − 1.46i)7-s + (−0.111 + 2.82i)8-s + (2.98 + 3.56i)9-s + (−2.28 − 1.10i)10-s + (3.87 − 1.03i)11-s + (−4.61 + 3.05i)12-s + (−2.49 + 1.16i)13-s + (2.05 + 1.21i)14-s + (−0.862 − 4.89i)15-s + (−1.81 − 3.56i)16-s + (−3.35 − 2.81i)17-s + ⋯ |
| L(s) = 1 | + (−0.872 + 0.488i)2-s + (−1.44 − 0.674i)3-s + (0.522 − 0.852i)4-s + (0.460 + 0.657i)5-s + (1.59 − 0.118i)6-s + (−0.318 − 0.552i)7-s + (−0.0393 + 0.999i)8-s + (0.996 + 1.18i)9-s + (−0.723 − 0.348i)10-s + (1.16 − 0.312i)11-s + (−1.33 + 0.881i)12-s + (−0.692 + 0.322i)13-s + (0.547 + 0.326i)14-s + (−0.222 − 1.26i)15-s + (−0.453 − 0.891i)16-s + (−0.812 − 0.682i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.161022 - 0.259702i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.161022 - 0.259702i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 - 0.690i)T \) |
| 19 | \( 1 + (4.34 - 0.309i)T \) |
| good | 3 | \( 1 + (2.50 + 1.16i)T + (1.92 + 2.29i)T^{2} \) |
| 5 | \( 1 + (-1.02 - 1.47i)T + (-1.71 + 4.69i)T^{2} \) |
| 7 | \( 1 + (0.843 + 1.46i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.87 + 1.03i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (2.49 - 1.16i)T + (8.35 - 9.95i)T^{2} \) |
| 17 | \( 1 + (3.35 + 2.81i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (1.39 + 7.88i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.0366 + 0.419i)T + (-28.5 - 5.03i)T^{2} \) |
| 31 | \( 1 + (3.32 + 5.75i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.13 + 2.13i)T + 37iT^{2} \) |
| 41 | \( 1 + (4.11 - 1.49i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (10.1 - 7.09i)T + (14.7 - 40.4i)T^{2} \) |
| 47 | \( 1 + (-1.06 - 1.27i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-7.23 + 10.3i)T + (-18.1 - 49.8i)T^{2} \) |
| 59 | \( 1 + (4.91 - 0.429i)T + (58.1 - 10.2i)T^{2} \) |
| 61 | \( 1 + (1.59 - 2.28i)T + (-20.8 - 57.3i)T^{2} \) |
| 67 | \( 1 + (-5.01 - 0.438i)T + (65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (1.56 + 0.276i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-2.12 - 5.84i)T + (-55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-2.64 + 0.963i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (8.68 + 2.32i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-4.81 - 1.75i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-7.60 + 9.06i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−11.27982298326803591460643520287, −10.51794328391367580962746651708, −9.745930073723658223827076135838, −8.505733193088096970537035138375, −6.97049764755124308121652956196, −6.71716488125677944609990317098, −6.01073545516010698775458560635, −4.61137517920271754677185875101, −2.09835059030955653514061642718, −0.34743891456957562665466808203,
1.67432814864699407788047160475, 3.79677733779727758256436086804, 5.03971617583706604804011456493, 6.12035368065446056168955807292, 7.02985867655900916444257006093, 8.730861750225327780517704328856, 9.356893175168097296456661578957, 10.19864077152418416657836288177, 10.97548483377505968516027117555, 11.99515911787242840428107066202
