| L(s) = 1 | + (−0.0597 + 0.184i)2-s + (1.06 + 0.226i)3-s + (1.58 + 1.15i)4-s + (−0.5 − 0.866i)5-s + (−0.105 + 0.182i)6-s + (1.34 − 0.598i)7-s + (−0.620 + 0.450i)8-s + (−1.65 − 0.736i)9-s + (0.189 − 0.0402i)10-s + (−0.181 + 1.73i)11-s + (1.43 + 1.59i)12-s + (−1.29 + 1.43i)13-s + (0.0297 + 0.283i)14-s + (−0.336 − 1.03i)15-s + (1.16 + 3.59i)16-s + (−0.775 − 7.38i)17-s + ⋯ |
| L(s) = 1 | + (−0.0422 + 0.130i)2-s + (0.615 + 0.130i)3-s + (0.793 + 0.576i)4-s + (−0.223 − 0.387i)5-s + (−0.0430 + 0.0746i)6-s + (0.508 − 0.226i)7-s + (−0.219 + 0.159i)8-s + (−0.551 − 0.245i)9-s + (0.0598 − 0.0127i)10-s + (−0.0548 + 0.522i)11-s + (0.413 + 0.459i)12-s + (−0.358 + 0.398i)13-s + (0.00795 + 0.0757i)14-s + (−0.0870 − 0.267i)15-s + (0.291 + 0.897i)16-s + (−0.188 − 1.79i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.40638 + 0.288053i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40638 + 0.288053i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (4.46 + 3.32i)T \) |
| good | 2 | \( 1 + (0.0597 - 0.184i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (-1.06 - 0.226i)T + (2.74 + 1.22i)T^{2} \) |
| 7 | \( 1 + (-1.34 + 0.598i)T + (4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (0.181 - 1.73i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (1.29 - 1.43i)T + (-1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (0.775 + 7.38i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (-1.75 - 1.95i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (0.928 - 0.674i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.996 - 3.06i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-3.20 + 5.54i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (8.17 - 1.73i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-0.233 - 0.259i)T + (-4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (2.62 + 8.06i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.04 - 0.465i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (-11.3 - 2.41i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 - 9.34T + 61T^{2} \) |
| 67 | \( 1 + (-1.40 - 2.44i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-11.1 - 4.95i)T + (47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (1.03 - 9.85i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (0.561 + 5.34i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (3.38 - 0.720i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (-0.383 - 0.278i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-13.3 - 9.71i)T + (29.9 + 92.2i)T^{2} \) |
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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
−12.98193506589123124683081749479, −11.78828677316714152461213111766, −11.39895956725467818535566365327, −9.785042858130945081843347940721, −8.774886228965812003845824357773, −7.76032995712766738281729994754, −6.97128537846834027554516103575, −5.27497512537532547611048147105, −3.73953313645788411182971903384, −2.33824918950660295274082116138,
2.04468445161275820914420943179, 3.32946119657864547700072399051, 5.35009347530205395523410796182, 6.46121705980213057285896582091, 7.78695300314087768244260253622, 8.584030086356989401199826026251, 10.01770642364732902170309853798, 10.98044880949852995773966532068, 11.62313710686653915318664975401, 12.90744398568381730945678053454
