| L(s) = 1 | + (0.642 − 0.766i)2-s + (−0.726 − 1.57i)3-s + (−0.173 − 0.984i)4-s + (1.96 − 1.06i)5-s + (−1.67 − 0.454i)6-s + (4.83 + 0.853i)7-s + (−0.866 − 0.500i)8-s + (−1.94 + 2.28i)9-s + (0.445 − 2.19i)10-s + (−1.60 + 0.584i)11-s + (−1.42 + 0.988i)12-s + (−1.95 − 2.32i)13-s + (3.76 − 3.15i)14-s + (−3.10 − 2.31i)15-s + (−0.939 + 0.342i)16-s + (−3.04 + 1.75i)17-s + ⋯ |
| L(s) = 1 | + (0.454 − 0.541i)2-s + (−0.419 − 0.907i)3-s + (−0.0868 − 0.492i)4-s + (0.878 − 0.477i)5-s + (−0.682 − 0.185i)6-s + (1.82 + 0.322i)7-s + (−0.306 − 0.176i)8-s + (−0.648 + 0.761i)9-s + (0.140 − 0.692i)10-s + (−0.484 + 0.176i)11-s + (−0.410 + 0.285i)12-s + (−0.541 − 0.645i)13-s + (1.00 − 0.843i)14-s + (−0.801 − 0.597i)15-s + (−0.234 + 0.0855i)16-s + (−0.737 + 0.425i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.181 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.08351 - 1.30154i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08351 - 1.30154i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.642 + 0.766i)T \) |
| 3 | \( 1 + (0.726 + 1.57i)T \) |
| 5 | \( 1 + (-1.96 + 1.06i)T \) |
| good | 7 | \( 1 + (-4.83 - 0.853i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (1.60 - 0.584i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.95 + 2.32i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (3.04 - 1.75i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.936 - 1.62i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.79 - 0.492i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.46 - 2.06i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.994 - 5.64i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.92 + 1.11i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.451 - 0.379i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.99 - 10.9i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (13.0 + 2.30i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 13.5iT - 53T^{2} \) |
| 59 | \( 1 + (-4.14 - 1.50i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.58 + 8.98i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-4.47 - 5.32i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.54 - 4.40i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (11.3 + 6.52i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.76 - 3.99i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.47 + 2.94i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (0.280 - 0.485i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.45 - 6.75i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−11.74681969949951846884947748109, −10.96079420562879778980714507673, −10.07401859483918537573211906946, −8.557447589210433716741693146032, −7.918275396450432170962768751214, −6.41838858677418896838976120219, −5.28506840135424408175060368550, −4.81217061360378970995451661530, −2.37876783880269770556079623381, −1.50013571180351682427847651803,
2.38750218519510380805570780994, 4.28892028757515149407215648500, 4.97568058997454717248750180844, 5.94517965415082109422250890286, 7.12684662808173551610852020582, 8.304412464106792180210164281979, 9.350465171528011574602884771641, 10.45795191394607768829103496869, 11.20901384389406478858217828923, 11.92678138025859801739322230710
