| L(s) = 1 | + (0.965 + 0.258i)2-s + (−1.73 + 0.0263i)3-s + (0.866 + 0.499i)4-s + (0.707 − 0.707i)5-s + (−1.67 − 0.422i)6-s + (0.431 + 1.60i)7-s + (0.707 + 0.707i)8-s + (2.99 − 0.0912i)9-s + (0.866 − 0.500i)10-s + (−0.985 + 3.67i)11-s + (−1.51 − 0.843i)12-s + (2.83 − 2.22i)13-s + 1.66i·14-s + (−1.20 + 1.24i)15-s + (0.500 + 0.866i)16-s + (−0.660 + 1.14i)17-s + ⋯ |
| L(s) = 1 | + (0.683 + 0.183i)2-s + (−0.999 + 0.0152i)3-s + (0.433 + 0.249i)4-s + (0.316 − 0.316i)5-s + (−0.685 − 0.172i)6-s + (0.162 + 0.608i)7-s + (0.249 + 0.249i)8-s + (0.999 − 0.0304i)9-s + (0.273 − 0.158i)10-s + (−0.297 + 1.10i)11-s + (−0.436 − 0.243i)12-s + (0.786 − 0.617i)13-s + 0.445i·14-s + (−0.311 + 0.321i)15-s + (0.125 + 0.216i)16-s + (−0.160 + 0.277i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.54736 + 0.582082i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.54736 + 0.582082i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.965 - 0.258i)T \) |
| 3 | \( 1 + (1.73 - 0.0263i)T \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
| 13 | \( 1 + (-2.83 + 2.22i)T \) |
| good | 7 | \( 1 + (-0.431 - 1.60i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (0.985 - 3.67i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (0.660 - 1.14i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.54 + 1.48i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-3.32 - 5.76i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.32 + 0.766i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-5.84 - 5.84i)T + 31iT^{2} \) |
| 37 | \( 1 + (9.72 + 2.60i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (4.33 + 1.16i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (6.30 + 3.63i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (9.18 + 9.18i)T + 47iT^{2} \) |
| 53 | \( 1 - 0.302iT - 53T^{2} \) |
| 59 | \( 1 + (-3.71 + 0.995i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-2.23 + 3.87i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.0586 - 0.218i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (3.95 + 14.7i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (0.682 - 0.682i)T - 73iT^{2} \) |
| 79 | \( 1 + 8.02T + 79T^{2} \) |
| 83 | \( 1 + (-8.80 + 8.80i)T - 83iT^{2} \) |
| 89 | \( 1 + (2.31 - 8.65i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (1.63 - 0.437i)T + (84.0 - 48.5i)T^{2} \) |
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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
−11.76348196126584737028325524624, −10.61422702133290379765301937138, −9.839735366144071553268898359529, −8.606718069076050568500684659194, −7.35428473007510332876730946294, −6.51065283769252768492039704378, −5.20514738858867513983526261442, −5.13623729045154012902615630134, −3.46538852088025783154350232025, −1.66328691577940210886919059580,
1.19435164930069443612617792224, 3.12496595940766236669326707509, 4.37766557748566531821644129267, 5.39134715353575286735158266490, 6.32573130104689755461402528086, 7.02017850731672811572841932594, 8.336602027112731707453387314173, 9.796450441516012263917106122074, 10.59448753068597068645608603231, 11.31200831353895994447185728220
