| L(s) = 1 | + (−1.62 − 1.36i)2-s + (−0.986 − 1.42i)3-s + (0.430 + 2.44i)4-s + (2.52 + 0.917i)5-s + (−0.338 + 3.65i)6-s + (0.168 − 0.957i)7-s + (0.508 − 0.880i)8-s + (−1.05 + 2.80i)9-s + (−2.83 − 4.91i)10-s + (0.297 − 0.108i)11-s + (3.05 − 3.02i)12-s + (−1.15 + 0.973i)13-s + (−1.57 + 1.32i)14-s + (−1.17 − 4.49i)15-s + (2.63 − 0.960i)16-s + (−0.587 − 1.01i)17-s + ⋯ |
| L(s) = 1 | + (−1.14 − 0.962i)2-s + (−0.569 − 0.822i)3-s + (0.215 + 1.22i)4-s + (1.12 + 0.410i)5-s + (−0.138 + 1.49i)6-s + (0.0638 − 0.361i)7-s + (0.179 − 0.311i)8-s + (−0.351 + 0.936i)9-s + (−0.897 − 1.55i)10-s + (0.0897 − 0.0326i)11-s + (0.881 − 0.872i)12-s + (−0.321 + 0.269i)13-s + (−0.421 + 0.353i)14-s + (−0.304 − 1.16i)15-s + (0.659 − 0.240i)16-s + (−0.142 − 0.246i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0136 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0136 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.290032 - 0.286089i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.290032 - 0.286089i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.986 + 1.42i)T \) |
| good | 2 | \( 1 + (1.62 + 1.36i)T + (0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (-2.52 - 0.917i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.168 + 0.957i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.297 + 0.108i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.15 - 0.973i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.587 + 1.01i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.11 - 5.38i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.375 - 2.12i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (3.37 + 2.83i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (1.50 + 8.54i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.23 - 3.86i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.47 + 3.75i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (5.25 - 1.91i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.429 - 2.43i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + (-1.62 - 0.589i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.176 + 0.999i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.656 + 0.550i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-4.79 - 8.31i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.62 + 13.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.59 + 7.20i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.58 - 3.01i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-7.74 + 13.4i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.21 + 1.89i)T + (74.3 - 62.3i)T^{2} \) |
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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
−17.38063797029706303578249134540, −16.85653595453129799690413926291, −14.34164920302392859617651024559, −13.07191318494187141752652377171, −11.70509465615510580674447160630, −10.60537146449475110024889059829, −9.551495980560636820671011895862, −7.78821171040463074569630825045, −6.04151511889640557530896434076, −2.02740572721175555232292198423,
5.30484352418663340812393184995, 6.60196378716140662785480752167, 8.754202135360183087992135881480, 9.556802985926483511356063025760, 10.74006234828547294047579888166, 12.71589541767922192295071025251, 14.63844241134237268771684156370, 15.69052381387282704267491922739, 16.77872370909022347335659810044, 17.46211532348221847348801431882
