| L(s) = 1 | + (−0.500 − 1.95i)2-s + (0.261 − 0.0499i)3-s + (−1.80 + 0.990i)4-s + (1.92 − 1.13i)5-s + (−0.228 − 0.485i)6-s + (0.923 − 1.27i)7-s + (0.0767 + 0.0816i)8-s + (−2.72 + 1.07i)9-s + (−3.17 − 3.19i)10-s + (−0.0932 + 0.0239i)11-s + (−0.421 + 0.348i)12-s + (0.949 + 2.39i)13-s + (−2.94 − 1.16i)14-s + (0.447 − 0.392i)15-s + (−2.08 + 3.28i)16-s + (1.73 − 3.15i)17-s + ⋯ |
| L(s) = 1 | + (−0.354 − 1.37i)2-s + (0.151 − 0.0288i)3-s + (−0.900 + 0.495i)4-s + (0.861 − 0.507i)5-s + (−0.0932 − 0.198i)6-s + (0.349 − 0.480i)7-s + (0.0271 + 0.0288i)8-s + (−0.907 + 0.359i)9-s + (−1.00 − 1.00i)10-s + (−0.0281 + 0.00721i)11-s + (−0.121 + 0.100i)12-s + (0.263 + 0.665i)13-s + (−0.786 − 0.311i)14-s + (0.115 − 0.101i)15-s + (−0.520 + 0.820i)16-s + (0.421 − 0.765i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.494457 - 0.886292i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.494457 - 0.886292i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.92 + 1.13i)T \) |
| good | 2 | \( 1 + (0.500 + 1.95i)T + (-1.75 + 0.963i)T^{2} \) |
| 3 | \( 1 + (-0.261 + 0.0499i)T + (2.78 - 1.10i)T^{2} \) |
| 7 | \( 1 + (-0.923 + 1.27i)T + (-2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (0.0932 - 0.0239i)T + (9.63 - 5.29i)T^{2} \) |
| 13 | \( 1 + (-0.949 - 2.39i)T + (-9.47 + 8.89i)T^{2} \) |
| 17 | \( 1 + (-1.73 + 3.15i)T + (-9.10 - 14.3i)T^{2} \) |
| 19 | \( 1 + (0.905 - 4.74i)T + (-17.6 - 6.99i)T^{2} \) |
| 23 | \( 1 + (-3.68 - 0.231i)T + (22.8 + 2.88i)T^{2} \) |
| 29 | \( 1 + (-6.92 - 0.874i)T + (28.0 + 7.21i)T^{2} \) |
| 31 | \( 1 + (4.99 + 2.74i)T + (16.6 + 26.1i)T^{2} \) |
| 37 | \( 1 + (-5.69 - 3.61i)T + (15.7 + 33.4i)T^{2} \) |
| 41 | \( 1 + (-0.510 - 8.10i)T + (-40.6 + 5.13i)T^{2} \) |
| 43 | \( 1 + (8.09 - 2.63i)T + (34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (1.33 - 1.42i)T + (-2.95 - 46.9i)T^{2} \) |
| 53 | \( 1 + (5.06 + 2.38i)T + (33.7 + 40.8i)T^{2} \) |
| 59 | \( 1 + (2.15 + 2.59i)T + (-11.0 + 57.9i)T^{2} \) |
| 61 | \( 1 + (-0.265 + 4.22i)T + (-60.5 - 7.64i)T^{2} \) |
| 67 | \( 1 + (0.743 + 5.88i)T + (-64.8 + 16.6i)T^{2} \) |
| 71 | \( 1 + (8.84 + 8.31i)T + (4.45 + 70.8i)T^{2} \) |
| 73 | \( 1 + (9.25 + 7.65i)T + (13.6 + 71.7i)T^{2} \) |
| 79 | \( 1 + (-3.13 - 16.4i)T + (-73.4 + 29.0i)T^{2} \) |
| 83 | \( 1 + (14.5 + 2.77i)T + (77.1 + 30.5i)T^{2} \) |
| 89 | \( 1 + (-5.98 + 7.22i)T + (-16.6 - 87.4i)T^{2} \) |
| 97 | \( 1 + (0.586 - 4.64i)T + (-93.9 - 24.1i)T^{2} \) |
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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
−12.91950031393140644085762339606, −11.81646148065030163887134329751, −11.01292221455309676324966817202, −9.990084482652270138188212173155, −9.162417315743530312814045643707, −8.116037119945394986187839444654, −6.24999795118429382517204244639, −4.71588547586603828440273085074, −2.98186549887915305603634142881, −1.50672962469626613109286839818,
2.81110472607444647264940138649, 5.30642752897381688528744133315, 6.05673757765522917017355138237, 7.12321453406268933252859941172, 8.472604526881369500681417296151, 9.044308476019612974240551218292, 10.42683758628062075012167361711, 11.60348692261422912015591151319, 13.07284012334492793566560377291, 14.20278839003480979128610339924
