| 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
−13.40728894522193764111189921570, −11.92860783185234690047098076739, −11.33162079264229322249138662603, −9.820005462847793395406391311899, −9.421039780478177572948022454288, −8.543371602305091114466379701845, −6.65232678390261161023222110885, −5.77442380022427856821658629171, −3.76546803634548218854731737251, −2.32591544755458193482415647584,
1.31638806573044909366716762733, 3.59730281093947056844424964193, 5.82757189405969757388967284386, 6.50935785626763260008888647461, 7.77715167808129174129333289184, 8.767438369790428296644626296118, 10.15407615092868050752754021164, 10.48243482962099233758537776050, 12.29740382020390322175952260135, 13.21954421151731880459361987173
