| L(s) = 1 | + (−1.72 − 0.192i)3-s + (0.0709 + 0.402i)5-s + (−2.76 − 2.31i)7-s + (2.92 + 0.661i)9-s + (0.933 − 5.29i)11-s + (−6.08 − 2.21i)13-s + (−0.0448 − 0.706i)15-s + (−1.42 − 2.47i)17-s + (−2.71 + 4.70i)19-s + (4.30 + 4.51i)21-s + (2.18 − 1.83i)23-s + (4.54 − 1.65i)25-s + (−4.90 − 1.70i)27-s + (2.41 − 0.877i)29-s + (−2.53 + 2.12i)31-s + ⋯ |
| L(s) = 1 | + (−0.993 − 0.110i)3-s + (0.0317 + 0.179i)5-s + (−1.04 − 0.875i)7-s + (0.975 + 0.220i)9-s + (0.281 − 1.59i)11-s + (−1.68 − 0.614i)13-s + (−0.0115 − 0.182i)15-s + (−0.346 − 0.600i)17-s + (−0.623 + 1.07i)19-s + (0.939 + 0.985i)21-s + (0.455 − 0.381i)23-s + (0.908 − 0.330i)25-s + (−0.944 − 0.327i)27-s + (0.447 − 0.162i)29-s + (−0.455 + 0.382i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.536 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.536 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.246843 - 0.449352i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.246843 - 0.449352i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.72 + 0.192i)T \) |
| good | 5 | \( 1 + (-0.0709 - 0.402i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (2.76 + 2.31i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.933 + 5.29i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (6.08 + 2.21i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.42 + 2.47i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.71 - 4.70i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.18 + 1.83i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-2.41 + 0.877i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.53 - 2.12i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (0.462 + 0.801i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.40 - 1.23i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.154 - 0.873i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-7.89 - 6.62i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 4.56T + 53T^{2} \) |
| 59 | \( 1 + (-1.06 - 6.02i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (4.86 + 4.07i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (5.49 + 2.00i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (5.61 + 9.72i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.59 + 6.21i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.0666 - 0.0242i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-15.4 + 5.63i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-3.42 + 5.93i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.10 + 11.9i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−12.10291051402012944885137373422, −10.80249809671940862784580763028, −10.37898300795435855862478741789, −9.258858321756024845981965408413, −7.69656200371766199302946173911, −6.73335287439559428324201633375, −5.90234233469836986497547011694, −4.57078567132512055744967946742, −3.09637719308880523199484070351, −0.46817638787955604420308824627,
2.31164225155612380105184480682, 4.37998610268395096103286725281, 5.25740727257972522616587462434, 6.64145104911304325569309451111, 7.16489758689805845332070603722, 9.162312091051405097577095060140, 9.632085764416041142493593948537, 10.69628754929308173660185231276, 11.98514293682877365550347198384, 12.44872457744219937922856672731
