| L(s) = 1 | + (−0.967 + 1.03i)2-s + (0.419 − 1.68i)3-s + (−0.128 − 1.99i)4-s + (−0.272 − 0.190i)5-s + (1.32 + 2.05i)6-s + (0.0188 + 0.0158i)7-s + (2.18 + 1.79i)8-s + (−2.64 − 1.41i)9-s + (0.460 − 0.0964i)10-s + (5.05 − 3.53i)11-s + (−3.40 − 0.622i)12-s + (−4.09 + 1.90i)13-s + (−0.0346 + 0.00415i)14-s + (−0.434 + 0.377i)15-s + (−3.96 + 0.511i)16-s + (4.74 − 2.73i)17-s + ⋯ |
| L(s) = 1 | + (−0.684 + 0.729i)2-s + (0.242 − 0.970i)3-s + (−0.0640 − 0.997i)4-s + (−0.121 − 0.0853i)5-s + (0.541 + 0.840i)6-s + (0.00713 + 0.00599i)7-s + (0.771 + 0.635i)8-s + (−0.882 − 0.470i)9-s + (0.145 − 0.0305i)10-s + (1.52 − 1.06i)11-s + (−0.983 − 0.179i)12-s + (−1.13 + 0.529i)13-s + (−0.00925 + 0.00110i)14-s + (−0.112 + 0.0975i)15-s + (−0.991 + 0.127i)16-s + (1.15 − 0.664i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.693285 - 0.587568i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.693285 - 0.587568i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.967 - 1.03i)T \) |
| 3 | \( 1 + (-0.419 + 1.68i)T \) |
| good | 5 | \( 1 + (0.272 + 0.190i)T + (1.71 + 4.69i)T^{2} \) |
| 7 | \( 1 + (-0.0188 - 0.0158i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-5.05 + 3.53i)T + (3.76 - 10.3i)T^{2} \) |
| 13 | \( 1 + (4.09 - 1.90i)T + (8.35 - 9.95i)T^{2} \) |
| 17 | \( 1 + (-4.74 + 2.73i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.88 - 0.505i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (2.05 + 2.45i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (5.08 + 2.36i)T + (18.6 + 22.2i)T^{2} \) |
| 31 | \( 1 + (5.78 + 6.89i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-2.01 + 7.52i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-3.60 - 1.31i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-7.12 - 10.1i)T + (-14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (-6.34 - 5.32i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (2.04 - 2.04i)T - 53iT^{2} \) |
| 59 | \( 1 + (5.56 - 7.94i)T + (-20.1 - 55.4i)T^{2} \) |
| 61 | \( 1 + (-4.89 + 0.427i)T + (60.0 - 10.5i)T^{2} \) |
| 67 | \( 1 + (0.836 + 1.79i)T + (-43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (-11.6 + 6.73i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.10 - 0.639i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.20 - 11.5i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-3.58 + 7.68i)T + (-53.3 - 63.5i)T^{2} \) |
| 89 | \( 1 + (0.858 - 1.48i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.562 + 3.18i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.05509406214958951044261645954, −9.518336999027202484788503897255, −9.160942392937809776889593944456, −7.979001391186141445258273349290, −7.43186539238005390503947132526, −6.35099489639640744584573042363, −5.74361539252996812367689213463, −4.08301037799773155546773123295, −2.24976509886778039642321034977, −0.71929062401352321200363509744,
1.86371615516524098166829134409, 3.38353737439106618403540155091, 4.12448725550407680419383352825, 5.36816860342710633078545978815, 7.06450293320908359737618029551, 7.87780896123628453870443180175, 9.064001733107597132855289561588, 9.577888040075902262897930983232, 10.30164586492087347688847647307, 11.14917025684856270453862900340
