| L(s) = 1 | + (−0.247 + 0.0391i)2-s + (0.786 − 1.54i)3-s + (−3.74 + 1.21i)4-s + (3.63 − 3.43i)5-s + (−0.133 + 0.411i)6-s + (8.31 − 8.31i)7-s + (1.76 − 0.901i)8-s + (−1.76 − 2.42i)9-s + (−0.763 + 0.990i)10-s + (2.05 + 1.49i)11-s + (−1.06 + 6.73i)12-s + (−16.0 − 2.54i)13-s + (−1.72 + 2.37i)14-s + (−2.43 − 8.31i)15-s + (12.3 − 8.96i)16-s + (14.9 + 29.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.123 + 0.0195i)2-s + (0.262 − 0.514i)3-s + (−0.936 + 0.304i)4-s + (0.727 − 0.686i)5-s + (−0.0223 + 0.0686i)6-s + (1.18 − 1.18i)7-s + (0.221 − 0.112i)8-s + (−0.195 − 0.269i)9-s + (−0.0763 + 0.0990i)10-s + (0.187 + 0.135i)11-s + (−0.0889 + 0.561i)12-s + (−1.23 − 0.195i)13-s + (−0.123 + 0.169i)14-s + (−0.162 − 0.554i)15-s + (0.771 − 0.560i)16-s + (0.879 + 1.72i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 + 0.824i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.09777 - 0.577553i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09777 - 0.577553i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.786 + 1.54i)T \) |
| 5 | \( 1 + (-3.63 + 3.43i)T \) |
| good | 2 | \( 1 + (0.247 - 0.0391i)T + (3.80 - 1.23i)T^{2} \) |
| 7 | \( 1 + (-8.31 + 8.31i)T - 49iT^{2} \) |
| 11 | \( 1 + (-2.05 - 1.49i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (16.0 + 2.54i)T + (160. + 52.2i)T^{2} \) |
| 17 | \( 1 + (-14.9 - 29.3i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (6.73 + 2.18i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + (-3.52 - 22.2i)T + (-503. + 163. i)T^{2} \) |
| 29 | \( 1 + (14.9 - 4.87i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (9.97 - 30.7i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (-0.828 + 5.22i)T + (-1.30e3 - 423. i)T^{2} \) |
| 41 | \( 1 + (-46.5 + 33.8i)T + (519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-38.4 - 38.4i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (6.82 + 3.47i)T + (1.29e3 + 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-0.196 + 0.384i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-6.75 - 9.29i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (53.2 + 38.7i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-7.76 - 15.2i)T + (-2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (-21.3 - 65.8i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-7.28 - 46.0i)T + (-5.06e3 + 1.64e3i)T^{2} \) |
| 79 | \( 1 + (53.4 - 17.3i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-72.7 + 37.0i)T + (4.04e3 - 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-9.87 + 13.5i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (67.0 + 34.1i)T + (5.53e3 + 7.61e3i)T^{2} \) |
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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
−14.15752238103665308025166070742, −13.04786852292929874964956383038, −12.36325051012927023941205259578, −10.59868075526488122308663717534, −9.491264461218375456687466367105, −8.267147609448663389221901810939, −7.45898248875085132282341678710, −5.38622996934403351234946560703, −4.13706486291951498376553338643, −1.38216156207686148492720162726,
2.46718059011044522526339218933, 4.74544089256128159565838901924, 5.65889287847631196692955816352, 7.72578208487649881659579320415, 9.110021287412146677466042300664, 9.698446573268029672548722621386, 10.99236252300019111522169071284, 12.21767117941917154214510167860, 13.79336849241816952016947402994, 14.60220779397953175994064986333
