| L(s) = 1 | + (0.651 + 1.25i)2-s + (1.08 − 1.35i)3-s + (−1.15 + 1.63i)4-s + (0.269 + 0.307i)5-s + (2.40 + 0.480i)6-s + (4.59 + 0.604i)7-s + (−2.80 − 0.380i)8-s + (−0.651 − 2.92i)9-s + (−0.210 + 0.538i)10-s + (−0.227 − 0.461i)11-s + (0.961 + 3.32i)12-s + (−1.02 − 3.02i)13-s + (2.23 + 6.15i)14-s + (0.707 − 0.0311i)15-s + (−1.34 − 3.76i)16-s + (5.43 + 5.43i)17-s + ⋯ |
| L(s) = 1 | + (0.460 + 0.887i)2-s + (0.625 − 0.780i)3-s + (−0.575 + 0.817i)4-s + (0.120 + 0.137i)5-s + (0.980 + 0.196i)6-s + (1.73 + 0.228i)7-s + (−0.990 − 0.134i)8-s + (−0.217 − 0.976i)9-s + (−0.0664 + 0.170i)10-s + (−0.0686 − 0.139i)11-s + (0.277 + 0.960i)12-s + (−0.284 − 0.838i)13-s + (0.596 + 1.64i)14-s + (0.182 − 0.00803i)15-s + (−0.337 − 0.941i)16-s + (1.31 + 1.31i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.29839 + 0.870013i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.29839 + 0.870013i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.651 - 1.25i)T \) |
| 3 | \( 1 + (-1.08 + 1.35i)T \) |
| good | 5 | \( 1 + (-0.269 - 0.307i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (-4.59 - 0.604i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (0.227 + 0.461i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (1.02 + 3.02i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (-5.43 - 5.43i)T + 17iT^{2} \) |
| 19 | \( 1 + (0.919 - 4.62i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-0.0161 - 0.122i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (4.85 - 0.318i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (-2.27 + 1.31i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.148 - 0.747i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-0.457 - 3.47i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (-2.07 + 1.02i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (8.80 - 2.36i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (4.67 - 6.99i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (4.63 + 5.28i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (-0.216 - 3.29i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (4.63 + 2.28i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (1.65 + 3.99i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-0.479 + 1.15i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (3.00 + 11.2i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-6.07 - 5.32i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (-1.71 + 0.709i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (12.8 + 7.39i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.92903766343018544110446952475, −9.743013917148541909339965659120, −8.418218201706435157326759946487, −8.055528462899566605396136090691, −7.56709445804149303605252105230, −6.17000600307405338422744383221, −5.53971052833449609548274613462, −4.29809220975063726960609511221, −3.09618072383565008871039729626, −1.61893110951053651805176243740,
1.56319828184189524892935489981, 2.70554151470888957973931002314, 3.97908517484631469791296351117, 4.94471555247706095416252699710, 5.26207835977096584245584279539, 7.22829684351950105892524239167, 8.209653895439752688943457950926, 9.190161672255591928914930064662, 9.716120093622195785032403314557, 10.83255402032348093344284408300
