| L(s) = 1 | + (0.0683 + 1.41i)2-s + (1.72 + 0.181i)3-s + (−1.99 + 0.193i)4-s + (0.342 − 0.939i)5-s + (−0.139 + 2.44i)6-s + (1.67 + 0.294i)7-s + (−0.408 − 2.79i)8-s + (2.93 + 0.626i)9-s + (1.35 + 0.418i)10-s + (3.70 − 1.34i)11-s + (−3.46 − 0.0294i)12-s + (−0.0310 + 0.0260i)13-s + (−0.301 + 2.37i)14-s + (0.759 − 1.55i)15-s + (3.92 − 0.768i)16-s + (0.268 − 0.154i)17-s + ⋯ |
| L(s) = 1 | + (0.0483 + 0.998i)2-s + (0.994 + 0.104i)3-s + (−0.995 + 0.0965i)4-s + (0.152 − 0.420i)5-s + (−0.0568 + 0.998i)6-s + (0.631 + 0.111i)7-s + (−0.144 − 0.989i)8-s + (0.977 + 0.208i)9-s + (0.427 + 0.132i)10-s + (1.11 − 0.406i)11-s + (−0.999 − 0.00849i)12-s + (−0.00860 + 0.00721i)13-s + (−0.0806 + 0.635i)14-s + (0.196 − 0.401i)15-s + (0.981 − 0.192i)16-s + (0.0650 − 0.0375i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.79078 + 1.12178i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.79078 + 1.12178i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0683 - 1.41i)T \) |
| 3 | \( 1 + (-1.72 - 0.181i)T \) |
| 5 | \( 1 + (-0.342 + 0.939i)T \) |
| good | 7 | \( 1 + (-1.67 - 0.294i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-3.70 + 1.34i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.0310 - 0.0260i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.268 + 0.154i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.04 + 1.75i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.533 - 3.02i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (4.25 - 5.07i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (6.28 - 1.10i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.75 - 4.76i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.39 - 8.80i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (4.16 + 11.4i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.963 + 5.46i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 2.69iT - 53T^{2} \) |
| 59 | \( 1 + (-0.0398 - 0.0145i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.476 + 2.70i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (8.20 + 9.77i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (0.821 + 1.42i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.41 - 4.18i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.72 + 3.24i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (0.385 + 0.323i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (7.73 + 4.46i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.108 + 0.0396i)T + (74.3 - 62.3i)T^{2} \) |
| 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
−10.81792523322766168575159858531, −9.567021100405258974599304972791, −9.006389655967449622692502113395, −8.379401094254515800809726841226, −7.47546518064304646070325256402, −6.57843671839249130228172084897, −5.36438146880707599083473165711, −4.38819251323502433914989020659, −3.44848864059912774562775934946, −1.55378248504710441571089209991,
1.54194519863626373489001371905, 2.49025403030089447967127841230, 3.81753606631719355514694150437, 4.43146466685137522345297593472, 6.00131554287280728565653944419, 7.31355281401260899820150143393, 8.190640848817820699416302144546, 9.133251214053358957048088720131, 9.693835117650589902494702225546, 10.69754992683501144263374548295
