| L(s) = 1 | + (−0.766 + 0.642i)2-s + (−1.67 − 0.453i)3-s + (0.173 − 0.984i)4-s + (1.96 − 0.346i)5-s + (1.57 − 0.726i)6-s + (0.910 − 1.57i)7-s + (0.500 + 0.866i)8-s + (2.58 + 1.51i)9-s + (−1.28 + 1.52i)10-s + (4.10 − 2.37i)11-s + (−0.737 + 1.56i)12-s + (0.151 + 0.415i)13-s + (0.316 + 1.79i)14-s + (−3.44 − 0.312i)15-s + (−0.939 − 0.342i)16-s + (1.07 + 1.28i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (−0.965 − 0.262i)3-s + (0.0868 − 0.492i)4-s + (0.879 − 0.155i)5-s + (0.641 − 0.296i)6-s + (0.344 − 0.596i)7-s + (0.176 + 0.306i)8-s + (0.862 + 0.505i)9-s + (−0.405 + 0.483i)10-s + (1.23 − 0.715i)11-s + (−0.212 + 0.452i)12-s + (0.0419 + 0.115i)13-s + (0.0845 + 0.479i)14-s + (−0.889 − 0.0808i)15-s + (−0.234 − 0.0855i)16-s + (0.260 + 0.310i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.744336 - 0.0648965i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.744336 - 0.0648965i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 3 | \( 1 + (1.67 + 0.453i)T \) |
| 19 | \( 1 + (3.58 + 2.48i)T \) |
| good | 5 | \( 1 + (-1.96 + 0.346i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.910 + 1.57i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.10 + 2.37i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.151 - 0.415i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.07 - 1.28i)T + (-2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (5.93 + 1.04i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-4.91 - 4.12i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-4.88 - 2.82i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 5.80iT - 37T^{2} \) |
| 41 | \( 1 + (3.75 + 1.36i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-2.15 - 12.2i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (6.92 - 8.25i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (0.424 - 2.40i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (3.87 - 3.24i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.80 + 10.2i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-5.27 + 6.28i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (0.897 + 5.08i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (13.5 + 4.94i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (3.23 - 8.88i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.523 - 0.302i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (4.07 - 1.48i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-1.64 - 1.96i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−13.73575737899069720863756930759, −12.46212873800162893455004796330, −11.28027392285962456829777006117, −10.42676451332973255052473350238, −9.371933659887288954628112392381, −8.067946957599470079048990167648, −6.61587370671602549906090415928, −6.01105605947656544194956839422, −4.51531602665346267929442897782, −1.40020863190698779221597899941,
1.87347750818937264734849717211, 4.21694370892189993686889567579, 5.78302773660402912611572062459, 6.76905794952695104701278108986, 8.472215440750695124223502175470, 9.839467865485084342046742348694, 10.17363570054528470602553100853, 11.84463572652728143173965460059, 11.95033564471553514524381767084, 13.44004216827065698652388378864
