| L(s) = 1 | + (0.664 + 1.24i)2-s + (−0.463 − 3.22i)3-s + (−1.11 + 1.65i)4-s + (0.304 + 0.473i)5-s + (3.71 − 2.71i)6-s + (2.18 + 3.40i)7-s + (−2.81 − 0.295i)8-s + (−7.29 + 2.14i)9-s + (−0.389 + 0.694i)10-s + (−1.48 + 2.30i)11-s + (5.86 + 2.83i)12-s + (0.728 + 5.06i)13-s + (−2.79 + 4.99i)14-s + (1.38 − 1.20i)15-s + (−1.49 − 3.70i)16-s + (3.01 − 3.47i)17-s + ⋯ |
| L(s) = 1 | + (0.469 + 0.882i)2-s + (−0.267 − 1.86i)3-s + (−0.559 + 0.829i)4-s + (0.136 + 0.211i)5-s + (1.51 − 1.10i)6-s + (0.826 + 1.28i)7-s + (−0.994 − 0.104i)8-s + (−2.43 + 0.713i)9-s + (−0.123 + 0.219i)10-s + (−0.447 + 0.695i)11-s + (1.69 + 0.818i)12-s + (0.202 + 1.40i)13-s + (−0.747 + 1.33i)14-s + (0.357 − 0.309i)15-s + (−0.374 − 0.927i)16-s + (0.730 − 0.842i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0692 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0692 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.943003 + 1.01073i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.943003 + 1.01073i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.664 - 1.24i)T \) |
| 89 | \( 1 + (5.43 + 7.71i)T \) |
| good | 3 | \( 1 + (0.463 + 3.22i)T + (-2.87 + 0.845i)T^{2} \) |
| 5 | \( 1 + (-0.304 - 0.473i)T + (-2.07 + 4.54i)T^{2} \) |
| 7 | \( 1 + (-2.18 - 3.40i)T + (-2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (1.48 - 2.30i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-0.728 - 5.06i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-3.01 + 3.47i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (7.00 - 2.05i)T + (15.9 - 10.2i)T^{2} \) |
| 23 | \( 1 + (-1.25 - 4.25i)T + (-19.3 + 12.4i)T^{2} \) |
| 29 | \( 1 + (2.93 - 1.88i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (1.52 - 5.18i)T + (-26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + (-2.25 - 0.323i)T + (39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-4.19 - 2.69i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (-0.118 + 0.822i)T + (-45.0 - 13.2i)T^{2} \) |
| 53 | \( 1 + (2.43 - 0.349i)T + (50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-0.178 + 1.23i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (0.849 + 1.85i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-9.15 + 1.31i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (10.7 + 6.92i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-0.188 - 0.0552i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (2.70 + 0.794i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (0.374 - 0.432i)T + (-11.8 - 82.1i)T^{2} \) |
| 97 | \( 1 + (11.6 - 7.45i)T + (40.2 - 88.2i)T^{2} \) |
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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
−11.16484242283004716015031752572, −9.308173537222914659589883209265, −8.528100903762602026344802317954, −7.81820424544504256165425438769, −7.07877355565415259695331952526, −6.28950791927689791208892701362, −5.61245804717544682743771728719, −4.64741895554760799999195181220, −2.69127045122588169918480108674, −1.82295144386393498696549002223,
0.64041244083120360492743490662, 2.81956930183690984363141657571, 3.92253980813429406299641738271, 4.38294330115306394413357225969, 5.37409440413130162149655641948, 5.99676859695585140212051152613, 8.020063503009615696124875655909, 8.735964460695439742087740121256, 9.795884078117049984526396363422, 10.48367786348101277122824451361
