| L(s) = 1 | + (0.845 + 1.13i)2-s + (1.05 + 1.37i)3-s + (−0.571 + 1.91i)4-s + (0.974 + 0.520i)5-s + (−0.668 + 2.35i)6-s + (−1.27 − 0.253i)7-s + (−2.65 + 0.971i)8-s + (−0.779 + 2.89i)9-s + (0.232 + 1.54i)10-s + (2.45 − 2.01i)11-s + (−3.23 + 1.23i)12-s + (2.24 − 1.19i)13-s + (−0.790 − 1.66i)14-s + (0.310 + 1.88i)15-s + (−3.34 − 2.19i)16-s + (−5.66 + 2.34i)17-s + ⋯ |
| L(s) = 1 | + (0.597 + 0.801i)2-s + (0.608 + 0.793i)3-s + (−0.285 + 0.958i)4-s + (0.435 + 0.232i)5-s + (−0.272 + 0.962i)6-s + (−0.482 − 0.0959i)7-s + (−0.939 + 0.343i)8-s + (−0.259 + 0.965i)9-s + (0.0736 + 0.488i)10-s + (0.739 − 0.607i)11-s + (−0.934 + 0.356i)12-s + (0.621 − 0.332i)13-s + (−0.211 − 0.444i)14-s + (0.0802 + 0.487i)15-s + (−0.836 − 0.547i)16-s + (−1.37 + 0.569i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.940907 + 1.94699i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.940907 + 1.94699i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.845 - 1.13i)T \) |
| 3 | \( 1 + (-1.05 - 1.37i)T \) |
| good | 5 | \( 1 + (-0.974 - 0.520i)T + (2.77 + 4.15i)T^{2} \) |
| 7 | \( 1 + (1.27 + 0.253i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-2.45 + 2.01i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-2.24 + 1.19i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (5.66 - 2.34i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-1.05 + 3.47i)T + (-15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (-7.00 - 4.68i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-2.10 - 1.72i)T + (5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (-6.37 + 6.37i)T - 31iT^{2} \) |
| 37 | \( 1 + (1.88 + 6.21i)T + (-30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (0.168 + 0.112i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (1.21 - 12.3i)T + (-42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (-3.43 + 1.42i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (3.10 - 2.54i)T + (10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (-1.79 - 0.958i)T + (32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (-0.0439 - 0.445i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-3.06 + 0.301i)T + (65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (-1.39 - 0.277i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-0.208 - 1.04i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (-4.58 + 11.0i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-2.49 + 8.22i)T + (-69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (-6.42 - 9.61i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (4.81 + 4.81i)T + 97iT^{2} \) |
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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
−11.54332430387607615534607658666, −10.81442507370671008903831561867, −9.483551429947513705368089284792, −8.933358545693665641937499494244, −8.008395803498195516332955020114, −6.72808334772066961354699181980, −5.96926283394507409207568743922, −4.73005538009103713798312265697, −3.72366473752303841015019105866, −2.74383140204487885403461476506,
1.31704561641498075738082312612, 2.51031185764484357317038114077, 3.69202213766360219140743207601, 4.94191711267061331948468042710, 6.39851969443091023275437302619, 6.82845133750858957589052398627, 8.618233967219985638869427030387, 9.182486271497311233635132665804, 10.05931438911023391408153487204, 11.26599347714938564897753809144
