| L(s) = 1 | + (1.13 − 4.24i)2-s + (−1.01 − 5.09i)3-s + (−9.83 − 5.67i)4-s + (8.66 + 8.66i)5-s + (−22.8 − 1.47i)6-s + (−1.76 + 0.472i)7-s + (−10.4 + 10.4i)8-s + (−24.9 + 10.3i)9-s + (46.6 − 26.9i)10-s + (39.3 + 10.5i)11-s + (−18.9 + 55.8i)12-s + (−21.4 − 41.6i)13-s + 8.02i·14-s + (35.3 − 52.9i)15-s + (−12.9 − 22.4i)16-s + (25.7 − 44.6i)17-s + ⋯ |
| L(s) = 1 | + (0.402 − 1.50i)2-s + (−0.195 − 0.980i)3-s + (−1.22 − 0.709i)4-s + (0.774 + 0.774i)5-s + (−1.55 − 0.100i)6-s + (−0.0951 + 0.0254i)7-s + (−0.461 + 0.461i)8-s + (−0.923 + 0.383i)9-s + (1.47 − 0.852i)10-s + (1.07 + 0.288i)11-s + (−0.455 + 1.34i)12-s + (−0.458 − 0.888i)13-s + 0.153i·14-s + (0.608 − 0.911i)15-s + (−0.202 − 0.350i)16-s + (0.367 − 0.636i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.797 + 0.603i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.797 + 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.492885 - 1.46815i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.492885 - 1.46815i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.01 + 5.09i)T \) |
| 13 | \( 1 + (21.4 + 41.6i)T \) |
| good | 2 | \( 1 + (-1.13 + 4.24i)T + (-6.92 - 4i)T^{2} \) |
| 5 | \( 1 + (-8.66 - 8.66i)T + 125iT^{2} \) |
| 7 | \( 1 + (1.76 - 0.472i)T + (297. - 171.5i)T^{2} \) |
| 11 | \( 1 + (-39.3 - 10.5i)T + (1.15e3 + 665.5i)T^{2} \) |
| 17 | \( 1 + (-25.7 + 44.6i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-41.1 - 153. i)T + (-5.94e3 + 3.42e3i)T^{2} \) |
| 23 | \( 1 + (-48.8 - 84.6i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (114. - 66.1i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (72.6 - 72.6i)T - 2.97e4iT^{2} \) |
| 37 | \( 1 + (-23.4 + 87.4i)T + (-4.38e4 - 2.53e4i)T^{2} \) |
| 41 | \( 1 + (15.6 - 58.5i)T + (-5.96e4 - 3.44e4i)T^{2} \) |
| 43 | \( 1 + (94.0 + 54.3i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-313. + 313. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 - 740. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (32.2 + 120. i)T + (-1.77e5 + 1.02e5i)T^{2} \) |
| 61 | \( 1 + (-217. + 376. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (730. + 195. i)T + (2.60e5 + 1.50e5i)T^{2} \) |
| 71 | \( 1 + (-406. + 108. i)T + (3.09e5 - 1.78e5i)T^{2} \) |
| 73 | \( 1 + (421. + 421. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + 123.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (176. + 176. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + (1.35e3 + 363. i)T + (6.10e5 + 3.52e5i)T^{2} \) |
| 97 | \( 1 + (21.3 + 79.8i)T + (-7.90e5 + 4.56e5i)T^{2} \) |
| 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
−14.50814358208588430815070898029, −13.84089129740070766272983328864, −12.61552861744446287215692195734, −11.85674866950934224520027235337, −10.62319151825116203392386869804, −9.537076827398364445849184984273, −7.31221590420705285214583604117, −5.67849859491527816197835343294, −3.17863532689783581581403871423, −1.60438524574507656145973593694,
4.36629203501569861211947545295, 5.49044197349190227332901309931, 6.72842469660735581448374922474, 8.755721177223856282714042830851, 9.491607708376805776939576285459, 11.40414570901855772230092561342, 13.16028749276983744457815805135, 14.26306132660202011651162440536, 15.09393483373420397443275472702, 16.34018684135774487413188302035
