| 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
−12.19503296346614910081866790010, −10.75897157872524164688128148214, −10.29202512757967340320297659885, −9.163370007160835186625255248338, −8.400627917554228966069469405719, −7.17413558209969294958578169532, −6.61003023280023731113731045780, −5.43144589505362531795455810071, −3.48975306911907353208217718376, −1.65515442341887576519238760989,
0.46269476705765809975497747092, 1.67477355304954389775118334037, 3.12342296557030860745274226432, 4.98699169795105973639077295679, 6.07741512636588565326304813934, 7.909221416250823095276586022290, 8.501990541660968049597941412966, 9.475458632552157647674853856152, 10.20158613117765368404895639815, 11.15702326248337415647069356587
