| L(s) = 1 | + (−0.341 + 1.37i)2-s + (0.841 − 0.540i)3-s + (−1.76 − 0.937i)4-s + (−0.116 − 0.255i)5-s + (0.454 + 1.33i)6-s + (−4.01 + 1.17i)7-s + (1.89 − 2.10i)8-s + (0.415 − 0.909i)9-s + (0.389 − 0.0727i)10-s + (3.52 − 3.05i)11-s + (−1.99 + 0.165i)12-s + (1.07 − 3.66i)13-s + (−0.245 − 5.90i)14-s + (−0.235 − 0.151i)15-s + (2.24 + 3.31i)16-s + (−6.59 − 0.948i)17-s + ⋯ |
| L(s) = 1 | + (−0.241 + 0.970i)2-s + (0.485 − 0.312i)3-s + (−0.883 − 0.468i)4-s + (−0.0521 − 0.114i)5-s + (0.185 + 0.546i)6-s + (−1.51 + 0.445i)7-s + (0.668 − 0.743i)8-s + (0.138 − 0.303i)9-s + (0.123 − 0.0229i)10-s + (1.06 − 0.921i)11-s + (−0.575 + 0.0479i)12-s + (0.298 − 1.01i)13-s + (−0.0656 − 1.57i)14-s + (−0.0609 − 0.0391i)15-s + (0.560 + 0.828i)16-s + (−1.59 − 0.230i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 + 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.978407 - 0.304697i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.978407 - 0.304697i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.341 - 1.37i)T \) |
| 3 | \( 1 + (-0.841 + 0.540i)T \) |
| 23 | \( 1 + (-4.34 + 2.03i)T \) |
| good | 5 | \( 1 + (0.116 + 0.255i)T + (-3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (4.01 - 1.17i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-3.52 + 3.05i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.07 + 3.66i)T + (-10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (6.59 + 0.948i)T + (16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-4.18 + 0.602i)T + (18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (4.89 + 0.703i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-3.52 + 5.48i)T + (-12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (1.02 - 2.25i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-0.284 - 0.622i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (5.63 + 8.76i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 0.838iT - 47T^{2} \) |
| 53 | \( 1 + (6.66 - 1.95i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-3.36 - 0.986i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-2.00 - 1.28i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-11.1 - 9.65i)T + (9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (9.29 + 8.05i)T + (10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.923 - 6.42i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-6.12 - 1.79i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (2.72 + 1.24i)T + (54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-2.40 - 3.74i)T + (-36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (14.2 - 6.49i)T + (63.5 - 73.3i)T^{2} \) |
| 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
−10.40315116833140453885143903721, −9.342666404386042869530642610280, −8.963974780250558344430651225796, −8.134246032540647710092925469592, −6.82992272978666473717127185549, −6.45210714470535124802153285687, −5.43923176984013633377748910226, −3.95198424065096094733200139386, −2.91400788490122564068619328172, −0.63758090868142763643127514330,
1.63873890163446148182402000890, 3.11266564659519282707869738144, 3.85125145534787183566660721766, 4.78533163156580841170775605094, 6.62322394713062884819539278336, 7.19518934208417096513915102059, 8.727248621482806088262625025192, 9.422614378612546789875510048250, 9.703185039376469721283120777647, 10.87155058936622706308193433920
