| L(s) = 1 | + (0.0523 + 0.998i)2-s + (−1.45 − 1.18i)3-s + (−0.994 + 0.104i)4-s + (−1.98 − 1.03i)5-s + (1.10 − 1.51i)6-s + (0.182 + 2.63i)7-s + (−0.156 − 0.987i)8-s + (0.109 + 0.515i)9-s + (0.932 − 2.03i)10-s + (5.35 + 1.13i)11-s + (1.57 + 1.02i)12-s + (2.02 + 3.97i)13-s + (−2.62 + 0.320i)14-s + (1.66 + 3.85i)15-s + (0.978 − 0.207i)16-s + (1.59 + 0.612i)17-s + ⋯ |
| L(s) = 1 | + (0.0370 + 0.706i)2-s + (−0.842 − 0.682i)3-s + (−0.497 + 0.0522i)4-s + (−0.885 − 0.464i)5-s + (0.450 − 0.620i)6-s + (0.0691 + 0.997i)7-s + (−0.0553 − 0.349i)8-s + (0.0365 + 0.171i)9-s + (0.294 − 0.642i)10-s + (1.61 + 0.342i)11-s + (0.454 + 0.295i)12-s + (0.561 + 1.10i)13-s + (−0.701 + 0.0857i)14-s + (0.429 + 0.995i)15-s + (0.244 − 0.0519i)16-s + (0.387 + 0.148i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.763610 + 0.427767i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.763610 + 0.427767i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0523 - 0.998i)T \) |
| 5 | \( 1 + (1.98 + 1.03i)T \) |
| 7 | \( 1 + (-0.182 - 2.63i)T \) |
| good | 3 | \( 1 + (1.45 + 1.18i)T + (0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (-5.35 - 1.13i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-2.02 - 3.97i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-1.59 - 0.612i)T + (12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (0.000953 - 0.00907i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-4.72 + 0.247i)T + (22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (-1.47 - 2.03i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (3.16 + 7.10i)T + (-20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (2.70 - 4.16i)T + (-15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (-4.63 - 1.50i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-5.48 - 5.48i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.82 + 7.36i)T + (-34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (6.66 - 8.22i)T + (-11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (-7.30 - 8.10i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (2.29 + 2.07i)T + (6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-2.96 + 7.71i)T + (-49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (-4.72 + 3.43i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (9.66 - 6.27i)T + (29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (-2.42 + 5.45i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-14.8 + 2.35i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-1.83 + 2.04i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (-14.8 - 2.35i)T + (92.2 + 29.9i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.79633211027495027810497190276, −11.21874854817299877651265539385, −9.233577646454955513234264656118, −8.946648756916195756389010150523, −7.66606188612989207419128675593, −6.68224215759424126759712739488, −6.07940927484201000317438197673, −4.86833973621051958577907764136, −3.75898690709365413567269764765, −1.29648886376238255454542446239,
0.837788760791178409790873756823, 3.38177677453793902581242168682, 4.03200988197061302915706635639, 5.13643849199985467110817193169, 6.43536153866113483689991312712, 7.55781134059754776040692751106, 8.694414537139781544315059759424, 9.876543619350294509301796618505, 10.88330928333222261121184970855, 10.99416096824099166065729933944
