L(s) = 1 | + 2-s + (−1.54 + 0.784i)3-s + 4-s + 3.23i·5-s + (−1.54 + 0.784i)6-s + (2.52 + 0.776i)7-s + 8-s + (1.76 − 2.42i)9-s + 3.23i·10-s + (3.60 − 2.07i)11-s + (−1.54 + 0.784i)12-s + (−2.79 − 1.61i)13-s + (2.52 + 0.776i)14-s + (−2.54 − 4.99i)15-s + 16-s + (5.17 + 2.98i)17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.891 + 0.453i)3-s + 0.5·4-s + 1.44i·5-s + (−0.630 + 0.320i)6-s + (0.955 + 0.293i)7-s + 0.353·8-s + (0.589 − 0.807i)9-s + 1.02i·10-s + (1.08 − 0.626i)11-s + (−0.445 + 0.226i)12-s + (−0.775 − 0.447i)13-s + (0.675 + 0.207i)14-s + (−0.656 − 1.29i)15-s + 0.250·16-s + (1.25 + 0.724i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0684 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0684 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.48477 + 1.38644i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.48477 + 1.38644i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (1.54 - 0.784i)T \) |
| 7 | \( 1 + (-2.52 - 0.776i)T \) |
| 19 | \( 1 + (0.893 - 4.26i)T \) |
good | 5 | \( 1 - 3.23iT - 5T^{2} \) |
| 11 | \( 1 + (-3.60 + 2.07i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.79 + 1.61i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-5.17 - 2.98i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (1.38 - 0.800i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.61 - 6.26i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.0503 + 0.0290i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (8.49 + 4.90i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.30 + 9.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.151 + 0.262i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.40 + 0.812i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 + (4.95 - 8.57i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.878 + 1.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + 6.48iT - 67T^{2} \) |
| 71 | \( 1 + (-7.51 - 13.0i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.01 + 1.76i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 1.11iT - 79T^{2} \) |
| 83 | \( 1 + 6.21iT - 83T^{2} \) |
| 89 | \( 1 + (3.98 + 6.89i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.4 + 7.78i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.51942421066089985120556568884, −10.20023775030415673340363097649, −8.800818617311857511622222024001, −7.52694303373487739020294169568, −6.86646871802544411593240792748, −5.78965334833050532100324746762, −5.42074535049325530791925185350, −3.95684602295392425515457993743, −3.35771070192908765241958898408, −1.73014909122294426857754393587,
1.00681913531655895225445759128, 1.98670755479152955878298989336, 4.10340850516305936240729630787, 4.89443084181088692393689823653, 5.21952074049537274227189863869, 6.50272885963996891420830563701, 7.34905727525107614395751250679, 8.135963850726519945845481040098, 9.303197406707465395567475356603, 10.12497419274842801111319609254