| L(s) = 1 | + (−0.104 − 0.994i)2-s + (2.12 + 2.35i)3-s + (−0.978 + 0.207i)4-s + (−1.19 + 1.89i)5-s + (2.12 − 2.35i)6-s − 4.64·7-s + (0.309 + 0.951i)8-s + (−0.736 + 7.00i)9-s + (2.00 + 0.990i)10-s + (1.47 − 1.07i)11-s + (−2.56 − 1.86i)12-s + (−0.0544 + 0.517i)13-s + (0.485 + 4.61i)14-s + (−6.98 + 1.19i)15-s + (0.913 − 0.406i)16-s + (0.146 + 0.0310i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 − 0.703i)2-s + (1.22 + 1.35i)3-s + (−0.489 + 0.103i)4-s + (−0.534 + 0.845i)5-s + (0.865 − 0.961i)6-s − 1.75·7-s + (0.109 + 0.336i)8-s + (−0.245 + 2.33i)9-s + (0.633 + 0.313i)10-s + (0.446 − 0.324i)11-s + (−0.740 − 0.537i)12-s + (−0.0150 + 0.143i)13-s + (0.129 + 1.23i)14-s + (−1.80 + 0.308i)15-s + (0.228 − 0.101i)16-s + (0.0354 + 0.00753i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.178327 + 0.900399i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.178327 + 0.900399i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 + 0.994i)T \) |
| 5 | \( 1 + (1.19 - 1.89i)T \) |
| 19 | \( 1 + (-0.547 + 4.32i)T \) |
| good | 3 | \( 1 + (-2.12 - 2.35i)T + (-0.313 + 2.98i)T^{2} \) |
| 7 | \( 1 + 4.64T + 7T^{2} \) |
| 11 | \( 1 + (-1.47 + 1.07i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (0.0544 - 0.517i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (-0.146 - 0.0310i)T + (15.5 + 6.91i)T^{2} \) |
| 23 | \( 1 + (2.60 + 1.16i)T + (15.3 + 17.0i)T^{2} \) |
| 29 | \( 1 + (9.74 - 2.07i)T + (26.4 - 11.7i)T^{2} \) |
| 31 | \( 1 + (-0.664 - 2.04i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-7.26 - 5.27i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-0.509 + 0.226i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (4.77 - 8.26i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (7.74 - 1.64i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-2.71 + 0.576i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.490 + 0.218i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (1.71 + 0.763i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (8.54 - 9.49i)T + (-7.00 - 66.6i)T^{2} \) |
| 71 | \( 1 + (-6.78 - 7.53i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (0.356 + 3.39i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (5.93 + 6.58i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-3.03 - 9.34i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-1.63 - 0.729i)T + (59.5 + 66.1i)T^{2} \) |
| 97 | \( 1 + (-3.96 - 4.40i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−10.15241321691119329603294732080, −9.652693158381494299988085352932, −9.113278731441272496425374947081, −8.229809501453786281495820380931, −7.18635369008943124446951902282, −6.12115847823948599191002826620, −4.61852959192006254330963288979, −3.69477016510864396662288002808, −3.23778522216592680893303440110, −2.51939434243501506458815410801,
0.36018421078678593408040273604, 1.87023581452063148210941252535, 3.38243116273908363070822968481, 3.94685553102138331900296045227, 5.76437082210453479936191317485, 6.42811146117180410373676162982, 7.39933706997899270080945434712, 7.77020842785337527012122775746, 8.760830531301984566516511211795, 9.338777864752595188056783684417
