| L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.596 + 3.38i)3-s + (0.766 + 0.642i)4-s + (0.532 − 0.447i)5-s + (0.596 − 3.38i)6-s + (−0.678 + 1.17i)7-s + (−0.500 − 0.866i)8-s + (−8.28 + 3.01i)9-s + (−0.653 + 0.237i)10-s + (0.5 + 0.866i)11-s + (−1.71 + 2.97i)12-s + (−0.418 + 2.37i)13-s + (1.03 − 0.872i)14-s + (1.83 + 1.53i)15-s + (0.173 + 0.984i)16-s + (−3.69 − 1.34i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (0.344 + 1.95i)3-s + (0.383 + 0.321i)4-s + (0.238 − 0.199i)5-s + (0.243 − 1.38i)6-s + (−0.256 + 0.444i)7-s + (−0.176 − 0.306i)8-s + (−2.76 + 1.00i)9-s + (−0.206 + 0.0752i)10-s + (0.150 + 0.261i)11-s + (−0.496 + 0.859i)12-s + (−0.115 + 0.657i)13-s + (0.277 − 0.233i)14-s + (0.472 + 0.396i)15-s + (0.0434 + 0.246i)16-s + (−0.895 − 0.326i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.195268 + 0.889305i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.195268 + 0.889305i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (-3.86 + 2.01i)T \) |
| good | 3 | \( 1 + (-0.596 - 3.38i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (-0.532 + 0.447i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (0.678 - 1.17i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (0.418 - 2.37i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (3.69 + 1.34i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-1.66 - 1.39i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-5.86 + 2.13i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (1.86 - 3.22i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 8.72T + 37T^{2} \) |
| 41 | \( 1 + (0.646 + 3.66i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-3.90 + 3.28i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (6.16 - 2.24i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-1.72 - 1.45i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-8.66 - 3.15i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-11.4 - 9.62i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (4.62 - 1.68i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-5.35 + 4.49i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.869 - 4.93i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (0.810 + 4.59i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.299 + 0.518i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.0595 - 0.337i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (5.89 + 2.14i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.31635612256155424036281281369, −10.45285206732724887189108927529, −9.658350063133202775629587132686, −9.076444528722064191192395088443, −8.592019952551016834253650311906, −7.07175723164259259726748564150, −5.58748499650812646536965189671, −4.69173176823826656668288089151, −3.55349361819363512887723726459, −2.46196925520241408699753389636,
0.67956035176261776126104033435, 2.04058546492651581171184526805, 3.20653478473536988848926360282, 5.55442037457847771940574801548, 6.57662720200653456157489050213, 7.00218011086869805717027273093, 8.090449540360596675349149199987, 8.548466108152492233944963185038, 9.764439270937271989169855138497, 10.88629725287768371433605009522
