| L(s) = 1 | + (0.961 + 1.03i)2-s + (−1.23 − 1.21i)3-s + (−0.151 + 1.99i)4-s + (0.0717 − 0.197i)5-s + (0.0764 − 2.44i)6-s + (−0.984 − 0.173i)7-s + (−2.21 + 1.75i)8-s + (0.0405 + 2.99i)9-s + (0.273 − 0.115i)10-s + (5.85 − 2.13i)11-s + (2.61 − 2.27i)12-s + (−3.71 + 3.11i)13-s + (−0.766 − 1.18i)14-s + (−0.328 + 0.155i)15-s + (−3.95 − 0.605i)16-s + (−4.41 + 2.54i)17-s + ⋯ |
| L(s) = 1 | + (0.679 + 0.733i)2-s + (−0.711 − 0.702i)3-s + (−0.0759 + 0.997i)4-s + (0.0320 − 0.0881i)5-s + (0.0312 − 0.999i)6-s + (−0.372 − 0.0656i)7-s + (−0.782 + 0.622i)8-s + (0.0135 + 0.999i)9-s + (0.0864 − 0.0363i)10-s + (1.76 − 0.642i)11-s + (0.754 − 0.656i)12-s + (−1.02 + 0.863i)13-s + (−0.204 − 0.317i)14-s + (−0.0847 + 0.0402i)15-s + (−0.988 − 0.151i)16-s + (−1.07 + 0.618i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.836831 + 1.16190i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.836831 + 1.16190i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.961 - 1.03i)T \) |
| 3 | \( 1 + (1.23 + 1.21i)T \) |
| 7 | \( 1 + (0.984 + 0.173i)T \) |
| good | 5 | \( 1 + (-0.0717 + 0.197i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (-5.85 + 2.13i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (3.71 - 3.11i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (4.41 - 2.54i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.39 - 3.11i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.34 - 7.64i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.785 + 0.936i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (6.62 - 1.16i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.533 - 0.923i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.152 - 0.181i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (1.44 + 3.96i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.716 - 4.06i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 11.4iT - 53T^{2} \) |
| 59 | \( 1 + (-2.24 - 0.817i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.920 - 5.21i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (0.575 + 0.686i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.384 - 0.666i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.12 + 7.13i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.20 + 10.9i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (9.07 + 7.61i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-6.48 - 3.74i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.22 + 1.53i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.99376471392207713630713585605, −9.428847842432198742785426068421, −8.881495210349307835294453109635, −7.52930775853260514336969603850, −7.01007922991519617715416548506, −6.23032932758856345453560817699, −5.47406087107550854199755880376, −4.38734042519010997140265229246, −3.35094368413679860130032949996, −1.62656551154805227177428850242,
0.66296254600582759759022547411, 2.53334713964873339990095631290, 3.66445016745850197793111898641, 4.65808292527447960835075614706, 5.21963700265015794510279027448, 6.57384729131981469065579531880, 6.87679325918316339451401092423, 8.940464648311848191217079313571, 9.511847020486846040629310262331, 10.15153876939031406193817261714
