| L(s) = 1 | + (−0.841 + 0.540i)2-s + (0.142 − 0.989i)3-s + (0.415 − 0.909i)4-s + (1.78 − 0.522i)5-s + (0.415 + 0.909i)6-s + (−2.54 − 2.93i)7-s + (0.142 + 0.989i)8-s + (−0.959 − 0.281i)9-s + (−1.21 + 1.40i)10-s + (1.81 + 1.16i)11-s + (−0.841 − 0.540i)12-s + (4.55 − 5.25i)13-s + (3.72 + 1.09i)14-s + (−0.264 − 1.83i)15-s + (−0.654 − 0.755i)16-s + (2.36 + 5.17i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 + 0.382i)2-s + (0.0821 − 0.571i)3-s + (0.207 − 0.454i)4-s + (0.796 − 0.233i)5-s + (0.169 + 0.371i)6-s + (−0.961 − 1.11i)7-s + (0.0503 + 0.349i)8-s + (−0.319 − 0.0939i)9-s + (−0.384 + 0.443i)10-s + (0.546 + 0.351i)11-s + (−0.242 − 0.156i)12-s + (1.26 − 1.45i)13-s + (0.996 + 0.292i)14-s + (−0.0681 − 0.474i)15-s + (−0.163 − 0.188i)16-s + (0.573 + 1.25i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.846097 - 0.324202i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.846097 - 0.324202i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.841 - 0.540i)T \) |
| 3 | \( 1 + (-0.142 + 0.989i)T \) |
| 23 | \( 1 + (4.40 + 1.89i)T \) |
| good | 5 | \( 1 + (-1.78 + 0.522i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (2.54 + 2.93i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-1.81 - 1.16i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-4.55 + 5.25i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.36 - 5.17i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (0.810 - 1.77i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.0455 + 0.0997i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-1.21 - 8.44i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-1.65 - 0.487i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (7.89 - 2.31i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.147 + 1.02i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 3.27T + 47T^{2} \) |
| 53 | \( 1 + (-7.20 - 8.31i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-1.62 + 1.88i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.899 - 6.25i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (3.07 - 1.97i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (0.478 - 0.307i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-5.79 + 12.6i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-9.59 + 11.0i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (9.91 + 2.91i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (0.730 - 5.07i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-6.38 + 1.87i)T + (81.6 - 52.4i)T^{2} \) |
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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
−13.21954421151731880459361987173, −12.29740382020390322175952260135, −10.48243482962099233758537776050, −10.15407615092868050752754021164, −8.767438369790428296644626296118, −7.77715167808129174129333289184, −6.50935785626763260008888647461, −5.82757189405969757388967284386, −3.59730281093947056844424964193, −1.31638806573044909366716762733,
2.32591544755458193482415647584, 3.76546803634548218854731737251, 5.77442380022427856821658629171, 6.65232678390261161023222110885, 8.543371602305091114466379701845, 9.421039780478177572948022454288, 9.820005462847793395406391311899, 11.33162079264229322249138662603, 11.92860783185234690047098076739, 13.40728894522193764111189921570
