| L(s) = 1 | + (−3.25 − 2.36i)2-s + (2.51 + 7.74i)4-s + (6.00 − 9.42i)5-s + 1.75·7-s + (0.174 − 0.537i)8-s + (−41.7 + 16.4i)10-s + (−18.9 − 13.7i)11-s + (43.1 − 31.3i)13-s + (−5.71 − 4.15i)14-s + (50.8 − 36.9i)16-s + (6.45 − 19.8i)17-s + (5.67 − 17.4i)19-s + (88.1 + 22.7i)20-s + (29.0 + 89.2i)22-s + (34.8 + 25.3i)23-s + ⋯ |
| L(s) = 1 | + (−1.14 − 0.834i)2-s + (0.314 + 0.967i)4-s + (0.537 − 0.843i)5-s + 0.0949·7-s + (0.00771 − 0.0237i)8-s + (−1.32 + 0.520i)10-s + (−0.518 − 0.376i)11-s + (0.919 − 0.668i)13-s + (−0.109 − 0.0792i)14-s + (0.794 − 0.577i)16-s + (0.0921 − 0.283i)17-s + (0.0685 − 0.211i)19-s + (0.985 + 0.254i)20-s + (0.281 + 0.865i)22-s + (0.316 + 0.229i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00348i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.00129567 + 0.743948i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00129567 + 0.743948i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-6.00 + 9.42i)T \) |
| good | 2 | \( 1 + (3.25 + 2.36i)T + (2.47 + 7.60i)T^{2} \) |
| 7 | \( 1 - 1.75T + 343T^{2} \) |
| 11 | \( 1 + (18.9 + 13.7i)T + (411. + 1.26e3i)T^{2} \) |
| 13 | \( 1 + (-43.1 + 31.3i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-6.45 + 19.8i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-5.67 + 17.4i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-34.8 - 25.3i)T + (3.75e3 + 1.15e4i)T^{2} \) |
| 29 | \( 1 + (41.2 + 126. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-74.8 + 230. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (288. - 209. i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (343. - 249. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 + 93.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-72.1 - 222. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-8.66 - 26.6i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-365. + 265. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (696. + 506. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + (191. - 590. i)T + (-2.43e5 - 1.76e5i)T^{2} \) |
| 71 | \( 1 + (292. + 898. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-556. - 404. i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (173. + 532. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-290. + 895. i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + (-912. - 663. i)T + (2.17e5 + 6.70e5i)T^{2} \) |
| 97 | \( 1 + (386. + 1.18e3i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−11.15702326248337415647069356587, −10.20158613117765368404895639815, −9.475458632552157647674853856152, −8.501990541660968049597941412966, −7.909221416250823095276586022290, −6.07741512636588565326304813934, −4.98699169795105973639077295679, −3.12342296557030860745274226432, −1.67477355304954389775118334037, −0.46269476705765809975497747092,
1.65515442341887576519238760989, 3.48975306911907353208217718376, 5.43144589505362531795455810071, 6.61003023280023731113731045780, 7.17413558209969294958578169532, 8.400627917554228966069469405719, 9.163370007160835186625255248338, 10.29202512757967340320297659885, 10.75897157872524164688128148214, 12.19503296346614910081866790010
