| L(s) = 1 | + (−0.909 − 1.08i)2-s + (−0.347 + 1.96i)4-s + (1.07 + 2.96i)5-s + (−0.250 − 1.42i)7-s + (2.44 − 1.41i)8-s + (2.23 − 3.86i)10-s + (−6.39 + 17.5i)11-s + (10.5 + 8.88i)13-s + (−1.31 + 1.56i)14-s + (−3.75 − 1.36i)16-s + (16.3 + 9.44i)17-s + (−1.14 − 1.98i)19-s + (−6.21 + 1.09i)20-s + (24.8 − 9.04i)22-s + (26.5 + 4.68i)23-s + ⋯ |
| L(s) = 1 | + (−0.454 − 0.541i)2-s + (−0.0868 + 0.492i)4-s + (0.215 + 0.592i)5-s + (−0.0357 − 0.202i)7-s + (0.306 − 0.176i)8-s + (0.223 − 0.386i)10-s + (−0.581 + 1.59i)11-s + (0.814 + 0.683i)13-s + (−0.0936 + 0.111i)14-s + (−0.234 − 0.0855i)16-s + (0.961 + 0.555i)17-s + (−0.0601 − 0.104i)19-s + (−0.310 + 0.0547i)20-s + (1.12 − 0.411i)22-s + (1.15 + 0.203i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.07502 + 0.344110i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07502 + 0.344110i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.909 + 1.08i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-1.07 - 2.96i)T + (-19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (0.250 + 1.42i)T + (-46.0 + 16.7i)T^{2} \) |
| 11 | \( 1 + (6.39 - 17.5i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (-10.5 - 8.88i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-16.3 - 9.44i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (1.14 + 1.98i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-26.5 - 4.68i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (17.2 + 20.5i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (4.55 - 25.8i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (33.8 - 58.6i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-13.0 + 15.5i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (32.2 + 11.7i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (46.4 - 8.19i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + 49.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-13.0 - 35.9i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (9.55 + 54.2i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (95.2 + 79.9i)T + (779. + 4.42e3i)T^{2} \) |
| 71 | \( 1 + (10.4 + 6.04i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-37.3 - 64.7i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-74.8 + 62.7i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (81.1 + 96.6i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (9.23 - 5.33i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-145. - 53.0i)T + (7.20e3 + 6.04e3i)T^{2} \) |
| 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
−12.62546154032159746083876721883, −11.64100497396552157395944792792, −10.51915328476116810623150752128, −9.966029600292839044344338663264, −8.774116995677183406778752014254, −7.52049516807923522588378630265, −6.59150147492782448439243426231, −4.84563932295120834000890578354, −3.34064857684358629812911680933, −1.77879391633676691165388897443,
0.889813721158858502650941304196, 3.24352790904497908483099158237, 5.26635447775279761640020232335, 5.89923692368723433301454077018, 7.44641734079402043085370186726, 8.515988399602807941937512589386, 9.149594768525861033470371637062, 10.51049439835131915422388259872, 11.29208618894894610237642029902, 12.77418777568104770048168195213
