| L(s) = 1 | + (0.841 − 0.540i)2-s + (−1.92 + 0.564i)3-s + (0.415 − 0.909i)4-s + (2.23 + 2.57i)5-s + (−1.31 + 1.51i)6-s + (3.63 − 2.33i)7-s + (−0.142 − 0.989i)8-s + (0.856 − 0.550i)9-s + (3.27 + 0.961i)10-s + (−0.829 − 0.957i)11-s + (−0.285 + 1.98i)12-s + (−0.620 + 4.31i)13-s + (1.79 − 3.93i)14-s + (−5.75 − 3.69i)15-s + (−0.654 − 0.755i)16-s + (−2.33 − 5.11i)17-s + ⋯ |
| L(s) = 1 | + (0.594 − 0.382i)2-s + (−1.11 + 0.326i)3-s + (0.207 − 0.454i)4-s + (0.999 + 1.15i)5-s + (−0.535 + 0.618i)6-s + (1.37 − 0.883i)7-s + (−0.0503 − 0.349i)8-s + (0.285 − 0.183i)9-s + (1.03 + 0.303i)10-s + (−0.250 − 0.288i)11-s + (−0.0823 + 0.572i)12-s + (−0.172 + 1.19i)13-s + (0.480 − 1.05i)14-s + (−1.48 − 0.954i)15-s + (−0.163 − 0.188i)16-s + (−0.566 − 1.24i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.28829 - 0.0190966i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28829 - 0.0190966i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.841 + 0.540i)T \) |
| 67 | \( 1 + (3.98 + 7.14i)T \) |
| good | 3 | \( 1 + (1.92 - 0.564i)T + (2.52 - 1.62i)T^{2} \) |
| 5 | \( 1 + (-2.23 - 2.57i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (-3.63 + 2.33i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.829 + 0.957i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.620 - 4.31i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (2.33 + 5.11i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.31 - 0.848i)T + (7.89 + 17.2i)T^{2} \) |
| 23 | \( 1 + (8.22 - 2.41i)T + (19.3 - 12.4i)T^{2} \) |
| 29 | \( 1 + 4.45T + 29T^{2} \) |
| 31 | \( 1 + (0.866 + 6.02i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 - 2.97T + 37T^{2} \) |
| 41 | \( 1 + (-2.27 - 4.98i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (2.44 + 5.36i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (1.94 - 0.572i)T + (39.5 - 25.4i)T^{2} \) |
| 53 | \( 1 + (-1.42 + 3.11i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-1.29 - 9.03i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (1.66 - 1.92i)T + (-8.68 - 60.3i)T^{2} \) |
| 71 | \( 1 + (2.14 - 4.69i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-7.10 + 8.20i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (2.16 - 15.0i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-0.501 - 0.579i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-4.65 - 1.36i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 - 5.57T + 97T^{2} \) |
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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
−13.68377597727478451778979570910, −11.64159886384719747834017177699, −11.37413815814694382875795048645, −10.51609707575858972326527528618, −9.647640146719492480889702013814, −7.52142809900011614548939332233, −6.36593908132803674389841632152, −5.36011228968060975511809142561, −4.23559764907363406539795147309, −2.12799096935702406407680807977,
1.86889633553571101712136318381, 4.74824687711239727481011761746, 5.50456975084884452682707772204, 6.08726196710391040064149698774, 7.932642369388946602623594578001, 8.805264693960891444652880836831, 10.40438437872712956853049232003, 11.53962822084012921304883916653, 12.49446298657270149953110974182, 12.89131784696591865154583822017
