| 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
−13.44004216827065698652388378864, −11.95033564471553514524381767084, −11.84463572652728143173965460059, −10.17363570054528470602553100853, −9.839467865485084342046742348694, −8.472215440750695124223502175470, −6.76905794952695104701278108986, −5.78302773660402912611572062459, −4.21694370892189993686889567579, −1.87347750818937264734849717211,
1.40020863190698779221597899941, 4.51531602665346267929442897782, 6.01105605947656544194956839422, 6.61587370671602549906090415928, 8.067946957599470079048990167648, 9.371933659887288954628112392381, 10.42676451332973255052473350238, 11.28027392285962456829777006117, 12.46212873800162893455004796330, 13.73575737899069720863756930759
