| L(s) = 1 | + (1.87 + 0.684i)2-s + (3.06 + 2.57i)4-s + (−0.203 + 1.15i)5-s + (19.9 − 16.7i)7-s + (4.00 + 6.92i)8-s + (−1.17 + 2.02i)10-s + (3.76 + 21.3i)11-s + (49.4 − 18.0i)13-s + (48.9 − 17.8i)14-s + (2.77 + 15.7i)16-s + (−6.38 + 11.0i)17-s + (20.4 + 35.3i)19-s + (−3.59 + 3.01i)20-s + (−7.52 + 42.6i)22-s + (100. + 84.2i)23-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (0.383 + 0.321i)4-s + (−0.0181 + 0.103i)5-s + (1.07 − 0.904i)7-s + (0.176 + 0.306i)8-s + (−0.0370 + 0.0641i)10-s + (0.103 + 0.584i)11-s + (1.05 − 0.384i)13-s + (0.935 − 0.340i)14-s + (0.0434 + 0.246i)16-s + (−0.0911 + 0.157i)17-s + (0.246 + 0.427i)19-s + (−0.0401 + 0.0336i)20-s + (−0.0728 + 0.413i)22-s + (0.909 + 0.763i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.79935 + 0.423523i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.79935 + 0.423523i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.87 - 0.684i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (0.203 - 1.15i)T + (-117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (-19.9 + 16.7i)T + (59.5 - 337. i)T^{2} \) |
| 11 | \( 1 + (-3.76 - 21.3i)T + (-1.25e3 + 455. i)T^{2} \) |
| 13 | \( 1 + (-49.4 + 18.0i)T + (1.68e3 - 1.41e3i)T^{2} \) |
| 17 | \( 1 + (6.38 - 11.0i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-20.4 - 35.3i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-100. - 84.2i)T + (2.11e3 + 1.19e4i)T^{2} \) |
| 29 | \( 1 + (123. + 45.1i)T + (1.86e4 + 1.56e4i)T^{2} \) |
| 31 | \( 1 + (257. + 215. i)T + (5.17e3 + 2.93e4i)T^{2} \) |
| 37 | \( 1 + (-90.5 + 156. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-132. + 48.2i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 + (-7.03 - 39.8i)T + (-7.47e4 + 2.71e4i)T^{2} \) |
| 47 | \( 1 + (404. - 339. i)T + (1.80e4 - 1.02e5i)T^{2} \) |
| 53 | \( 1 + 493.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-146. + 832. i)T + (-1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (651. - 546. i)T + (3.94e4 - 2.23e5i)T^{2} \) |
| 67 | \( 1 + (369. - 134. i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (-301. + 521. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-52.8 - 91.5i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (210. + 76.6i)T + (3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (817. + 297. i)T + (4.38e5 + 3.67e5i)T^{2} \) |
| 89 | \( 1 + (194. + 336. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-259. - 1.47e3i)T + (-8.57e5 + 3.12e5i)T^{2} \) |
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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
−12.71966082370621014628626632993, −11.19636740861247049815846808740, −10.95655641396898765426393963153, −9.372220332982352740638764278362, −7.939851697058775160783340996913, −7.25812867412003893265157949572, −5.84931190368751560087567278166, −4.64764397464974710618233064081, −3.54572749158278541777422435748, −1.52579608106516130348456429528,
1.50501964438379637905651477007, 3.09802460854290247930586664785, 4.66509157969554406136911143829, 5.60639011320909283036904621363, 6.84469651871253268822976506656, 8.396407908357733267829918915706, 9.102297130569708960835192652085, 10.88088517383543848490127607136, 11.29030562716536852219902051933, 12.35969484404873378937779838748
