| L(s) = 1 | + (−3.07 − 4.60i)3-s + (1.22 − 6.15i)5-s + (0.478 + 2.40i)7-s + (−8.29 + 20.0i)9-s + (−12.9 − 8.63i)11-s + (13.3 + 13.3i)13-s + (−32.1 + 13.3i)15-s + (−13.2 − 10.6i)17-s + (−5.91 − 14.2i)19-s + (9.60 − 9.60i)21-s + (−4.55 + 6.82i)23-s + (−13.2 − 5.49i)25-s + (68.9 − 13.7i)27-s + (7.84 + 1.56i)29-s + (22.3 − 14.9i)31-s + ⋯ |
| L(s) = 1 | + (−1.02 − 1.53i)3-s + (0.244 − 1.23i)5-s + (0.0683 + 0.343i)7-s + (−0.922 + 2.22i)9-s + (−1.17 − 0.785i)11-s + (1.02 + 1.02i)13-s + (−2.14 + 0.886i)15-s + (−0.778 − 0.628i)17-s + (−0.311 − 0.752i)19-s + (0.457 − 0.457i)21-s + (−0.198 + 0.296i)23-s + (−0.531 − 0.219i)25-s + (2.55 − 0.507i)27-s + (0.270 + 0.0538i)29-s + (0.719 − 0.480i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0582i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 - 0.0582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0208180 + 0.714416i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0208180 + 0.714416i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (13.2 + 10.6i)T \) |
| good | 3 | \( 1 + (3.07 + 4.60i)T + (-3.44 + 8.31i)T^{2} \) |
| 5 | \( 1 + (-1.22 + 6.15i)T + (-23.0 - 9.56i)T^{2} \) |
| 7 | \( 1 + (-0.478 - 2.40i)T + (-45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (12.9 + 8.63i)T + (46.3 + 111. i)T^{2} \) |
| 13 | \( 1 + (-13.3 - 13.3i)T + 169iT^{2} \) |
| 19 | \( 1 + (5.91 + 14.2i)T + (-255. + 255. i)T^{2} \) |
| 23 | \( 1 + (4.55 - 6.82i)T + (-202. - 488. i)T^{2} \) |
| 29 | \( 1 + (-7.84 - 1.56i)T + (776. + 321. i)T^{2} \) |
| 31 | \( 1 + (-22.3 + 14.9i)T + (367. - 887. i)T^{2} \) |
| 37 | \( 1 + (32.9 + 49.3i)T + (-523. + 1.26e3i)T^{2} \) |
| 41 | \( 1 + (7.55 + 37.9i)T + (-1.55e3 + 643. i)T^{2} \) |
| 43 | \( 1 + (15.2 - 36.9i)T + (-1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (3.87 + 3.87i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (29.7 + 71.9i)T + (-1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-30.3 - 12.5i)T + (2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (3.81 - 0.759i)T + (3.43e3 - 1.42e3i)T^{2} \) |
| 67 | \( 1 + 72.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (36.3 + 54.3i)T + (-1.92e3 + 4.65e3i)T^{2} \) |
| 73 | \( 1 + (11.8 - 59.3i)T + (-4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-75.3 - 50.3i)T + (2.38e3 + 5.76e3i)T^{2} \) |
| 83 | \( 1 + (-13.8 + 5.74i)T + (4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-78.9 + 78.9i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (132. + 26.4i)T + (8.69e3 + 3.60e3i)T^{2} \) |
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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
−12.53602602621488396595305434580, −11.60323711122467957300240768115, −10.88385634012532838840750639243, −8.958874100709962423802936820551, −8.182590049644766797875827072368, −6.85004102284832709102558627151, −5.80320558377420077968925208938, −4.92393966692558910480304190681, −2.05733263393899324831787943704, −0.53059120325680725268829438664,
3.15001386104285476591845860322, 4.45286101616805095588871579180, 5.70721456047894324120104987804, 6.65292820196414056892432444598, 8.335882996606682733428720573136, 10.01573904786195986625913544804, 10.50236168236946307268923294896, 10.89623301984105267400307921128, 12.22777382701155768016058016587, 13.57042802428501138072182953149
