| L(s) = 1 | + (−1.22 − 0.856i)2-s + (−1.44 + 0.960i)3-s + (0.0790 + 0.217i)4-s + (2.10 + 0.751i)5-s + (2.58 + 0.0591i)6-s + (−2.03 − 4.36i)7-s + (−0.683 + 2.55i)8-s + (1.15 − 2.76i)9-s + (−1.93 − 2.72i)10-s + (−2.25 − 2.68i)11-s + (−0.322 − 0.237i)12-s + (−2.18 − 3.11i)13-s + (−1.24 + 7.08i)14-s + (−3.75 + 0.940i)15-s + (3.37 − 2.83i)16-s + (−0.367 − 1.37i)17-s + ⋯ |
| L(s) = 1 | + (−0.865 − 0.605i)2-s + (−0.832 + 0.554i)3-s + (0.0395 + 0.108i)4-s + (0.941 + 0.336i)5-s + (1.05 + 0.0241i)6-s + (−0.769 − 1.65i)7-s + (−0.241 + 0.902i)8-s + (0.384 − 0.923i)9-s + (−0.611 − 0.861i)10-s + (−0.678 − 0.808i)11-s + (−0.0931 − 0.0684i)12-s + (−0.605 − 0.864i)13-s + (−0.333 + 1.89i)14-s + (−0.970 + 0.242i)15-s + (0.844 − 0.708i)16-s + (−0.0891 − 0.332i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.552 + 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.209053 - 0.389218i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.209053 - 0.389218i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.44 - 0.960i)T \) |
| 5 | \( 1 + (-2.10 - 0.751i)T \) |
| good | 2 | \( 1 + (1.22 + 0.856i)T + (0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (2.03 + 4.36i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (2.25 + 2.68i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (2.18 + 3.11i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (0.367 + 1.37i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.30 + 0.750i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.35 - 0.633i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (0.168 + 0.957i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.44 - 0.891i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-6.69 + 1.79i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (0.670 + 0.118i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.0175 - 0.200i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (7.89 - 3.68i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (-2.81 - 2.81i)T + 53iT^{2} \) |
| 59 | \( 1 + (-5.69 - 4.77i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.08 + 0.396i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-12.5 + 8.79i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (11.1 + 6.42i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.343 + 0.0920i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-9.66 + 1.70i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-6.51 + 9.30i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (-2.09 - 3.63i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.41 - 0.561i)T + (95.5 + 16.8i)T^{2} \) |
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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
−12.87309310447220393870300841959, −11.25218115997559787151504436886, −10.57225502143335961402772509149, −10.05201719856187033023279757175, −9.312717243223898322145925172027, −7.53443900699123125339637201891, −6.19928867247643370795496351900, −5.07564182306675690017528796047, −3.14138454009819792059573086498, −0.65174548838181377175803436618,
2.23797824045562869666834622090, 5.10796701003172046966929244417, 6.15215539600695234082545088488, 6.99900611996225882445074204498, 8.363619581543832011973128864123, 9.429012207850393120318511818402, 10.03042268412648054994845125573, 11.77550269078953250849766470943, 12.69887362373243967020733084090, 13.07435680624739674315046915435
