| L(s) = 1 | + (−0.577 + 1.23i)2-s + (−1.61 + 0.618i)3-s + (0.0838 + 0.0999i)4-s + (−1.26 − 1.84i)5-s + (0.169 − 2.36i)6-s + (−0.636 + 2.37i)7-s + (−2.81 + 0.753i)8-s + (2.23 − 2.00i)9-s + (3.01 − 0.505i)10-s + (−1.15 − 0.664i)11-s + (−0.197 − 0.109i)12-s + (−4.10 − 2.87i)13-s + (−2.57 − 2.16i)14-s + (3.18 + 2.19i)15-s + (0.646 − 3.66i)16-s + (1.40 − 3.01i)17-s + ⋯ |
| L(s) = 1 | + (−0.408 + 0.876i)2-s + (−0.934 + 0.356i)3-s + (0.0419 + 0.0499i)4-s + (−0.566 − 0.824i)5-s + (0.0690 − 0.964i)6-s + (−0.240 + 0.897i)7-s + (−0.994 + 0.266i)8-s + (0.745 − 0.666i)9-s + (0.953 − 0.159i)10-s + (−0.347 − 0.200i)11-s + (−0.0569 − 0.0317i)12-s + (−1.13 − 0.797i)13-s + (−0.688 − 0.577i)14-s + (0.823 + 0.567i)15-s + (0.161 − 0.916i)16-s + (0.341 − 0.732i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0366 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0366 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0583853 - 0.0562823i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0583853 - 0.0562823i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.61 - 0.618i)T \) |
| 5 | \( 1 + (1.26 + 1.84i)T \) |
| 19 | \( 1 + (2.95 + 3.20i)T \) |
| good | 2 | \( 1 + (0.577 - 1.23i)T + (-1.28 - 1.53i)T^{2} \) |
| 7 | \( 1 + (0.636 - 2.37i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (1.15 + 0.664i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (4.10 + 2.87i)T + (4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-1.40 + 3.01i)T + (-10.9 - 13.0i)T^{2} \) |
| 23 | \( 1 + (-5.12 - 0.447i)T + (22.6 + 3.99i)T^{2} \) |
| 29 | \( 1 + (-2.16 - 0.788i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.564 - 0.977i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (7.02 + 7.02i)T + 37iT^{2} \) |
| 41 | \( 1 + (11.5 + 2.03i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (2.03 - 0.177i)T + (42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (10.2 - 4.79i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (-4.09 - 0.358i)T + (52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (11.7 - 4.27i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (8.41 - 7.06i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-3.25 - 6.98i)T + (-43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (-3.28 + 3.90i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (0.142 + 0.203i)T + (-24.9 + 68.5i)T^{2} \) |
| 79 | \( 1 + (-6.77 - 1.19i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.58 + 5.91i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-0.552 - 3.13i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-5.12 - 2.39i)T + (62.3 + 74.3i)T^{2} \) |
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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
−11.88089019217088204919573901400, −10.67823673349885345310382176766, −9.400469249929610600717095652092, −8.739721427110769439592402912593, −7.61789503532010372883222948478, −6.74121125893439724803791954413, −5.46292315306189105468924640557, −4.94655859247077553407464309990, −3.05181897868739013095806047694, −0.07359964056350789409896136924,
1.82712731959374464464859422679, 3.41955778966388833770048479379, 4.83073408591576881251414667331, 6.48105982320652418152749247492, 6.93123089190094094970690857962, 8.122977351769839533440124343967, 9.884447805535136369323635200529, 10.34250156260163549320713387879, 11.02318288226971445467076024841, 11.92475194935921187330814650950
