| L(s) = 1 | + (1.40 + 0.129i)2-s + (1.96 + 0.363i)4-s + (−0.297 − 0.0525i)5-s + (0.312 + 0.371i)7-s + (2.72 + 0.766i)8-s + (−0.412 − 0.112i)10-s + (−0.00722 − 0.0410i)11-s + (4.05 − 1.47i)13-s + (0.391 + 0.564i)14-s + (3.73 + 1.43i)16-s + (2.14 + 1.23i)17-s + (−6.55 + 3.78i)19-s + (−0.566 − 0.211i)20-s + (−0.00488 − 0.0586i)22-s + (−5.03 − 4.22i)23-s + ⋯ |
| L(s) = 1 | + (0.995 + 0.0913i)2-s + (0.983 + 0.181i)4-s + (−0.133 − 0.0234i)5-s + (0.117 + 0.140i)7-s + (0.962 + 0.270i)8-s + (−0.130 − 0.0355i)10-s + (−0.00217 − 0.0123i)11-s + (1.12 − 0.409i)13-s + (0.104 + 0.150i)14-s + (0.933 + 0.357i)16-s + (0.520 + 0.300i)17-s + (−1.50 + 0.868i)19-s + (−0.126 − 0.0473i)20-s + (−0.00104 − 0.0125i)22-s + (−1.04 − 0.880i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.37586 + 0.251672i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.37586 + 0.251672i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 - 0.129i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (0.297 + 0.0525i)T + (4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.312 - 0.371i)T + (-1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.00722 + 0.0410i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-4.05 + 1.47i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.14 - 1.23i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (6.55 - 3.78i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.03 + 4.22i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-2.40 + 6.60i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.30 - 2.75i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (3.36 - 5.81i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.16 + 5.94i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-2.38 + 0.420i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (5.51 - 4.62i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 9.90iT - 53T^{2} \) |
| 59 | \( 1 + (-0.755 + 4.28i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (5.77 - 4.84i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (3.80 + 10.4i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-1.68 + 2.92i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.315 - 0.546i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.03 + 2.84i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-5.41 - 1.96i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (9.00 - 5.19i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.578 + 3.28i)T + (-91.1 + 33.1i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.95032832196232831703028076557, −10.78102352978655116815208878337, −10.19154931455697255750023462859, −8.461107875851463024674532754842, −7.88043891457568628996410294599, −6.37725299860448828351876708189, −5.86722095558309222126299529710, −4.43304150150144379313186463246, −3.54614104482374446288270829374, −1.97913791971568302098609393925,
1.83068120731176316758606531424, 3.43817611686146976033397375397, 4.36476699897026835491349705868, 5.60887949750283632846579837595, 6.53353463832685973781115292237, 7.54739605179952647717041929354, 8.675424433738206155070913804102, 9.972032926799380105407394776383, 11.01014126905445395932479425325, 11.54579286208359606798428482416
