| L(s) = 1 | + (−0.792 − 3.72i)2-s + (2.16 + 4.72i)3-s + (−5.96 + 2.65i)4-s + (9.75 − 5.46i)5-s + (15.8 − 11.8i)6-s + (−4.93 + 2.85i)7-s + (−3.29 − 4.53i)8-s + (−17.6 + 20.4i)9-s + (−28.1 − 32.0i)10-s + (−3.91 + 0.831i)11-s + (−25.4 − 22.4i)12-s + (18.8 − 88.8i)13-s + (14.5 + 16.1i)14-s + (46.9 + 34.2i)15-s + (−49.2 + 54.6i)16-s + (27.5 + 37.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.280 − 1.31i)2-s + (0.416 + 0.909i)3-s + (−0.745 + 0.332i)4-s + (0.872 − 0.489i)5-s + (1.08 − 0.803i)6-s + (−0.266 + 0.153i)7-s + (−0.145 − 0.200i)8-s + (−0.653 + 0.757i)9-s + (−0.889 − 1.01i)10-s + (−0.107 + 0.0227i)11-s + (−0.612 − 0.539i)12-s + (0.402 − 1.89i)13-s + (0.277 + 0.308i)14-s + (0.808 + 0.589i)15-s + (−0.769 + 0.854i)16-s + (0.393 + 0.541i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.403 + 0.914i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.403 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.02646 - 1.57442i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02646 - 1.57442i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.16 - 4.72i)T \) |
| 5 | \( 1 + (-9.75 + 5.46i)T \) |
| good | 2 | \( 1 + (0.792 + 3.72i)T + (-7.30 + 3.25i)T^{2} \) |
| 7 | \( 1 + (4.93 - 2.85i)T + (171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (3.91 - 0.831i)T + (1.21e3 - 541. i)T^{2} \) |
| 13 | \( 1 + (-18.8 + 88.8i)T + (-2.00e3 - 893. i)T^{2} \) |
| 17 | \( 1 + (-27.5 - 37.9i)T + (-1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-123. + 89.8i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-66.4 + 59.8i)T + (1.27e3 - 1.21e4i)T^{2} \) |
| 29 | \( 1 + (-3.36 + 32.0i)T + (-2.38e4 - 5.07e3i)T^{2} \) |
| 31 | \( 1 + (14.1 + 134. i)T + (-2.91e4 + 6.19e3i)T^{2} \) |
| 37 | \( 1 + (102. - 33.2i)T + (4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-161. - 34.4i)T + (6.29e4 + 2.80e4i)T^{2} \) |
| 43 | \( 1 + (-196. + 113. i)T + (3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (162. + 17.0i)T + (1.01e5 + 2.15e4i)T^{2} \) |
| 53 | \( 1 + (131. - 181. i)T + (-4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (870. + 184. i)T + (1.87e5 + 8.35e4i)T^{2} \) |
| 61 | \( 1 + (269. - 57.2i)T + (2.07e5 - 9.23e4i)T^{2} \) |
| 67 | \( 1 + (-614. + 64.5i)T + (2.94e5 - 6.25e4i)T^{2} \) |
| 71 | \( 1 + (-244. - 177. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-697. - 226. i)T + (3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (0.941 - 8.95i)T + (-4.82e5 - 1.02e5i)T^{2} \) |
| 83 | \( 1 + (-133. + 300. i)T + (-3.82e5 - 4.24e5i)T^{2} \) |
| 89 | \( 1 + (359. - 1.10e3i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-1.59e3 - 168. i)T + (8.92e5 + 1.89e5i)T^{2} \) |
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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
−11.11776003099810912930927426160, −10.48724721109665393683125860257, −9.670481066178255834125237910198, −9.103897178001869098353554465638, −7.980688088520708741636596455700, −5.96700651650016136905506713319, −4.94004860238223245679751195580, −3.36442916163239176738897312887, −2.58642958607601037547135616036, −0.858399105846531899467876932063,
1.62017821426440917249366366551, 3.18401145128578393208895243544, 5.36294433365906649526743442133, 6.36662825425136592995861652970, 6.99342641411278294991750025099, 7.78731640034444037058675798356, 9.088049358419175744140882178627, 9.557646896239606294329781134055, 11.25827390844794899916805170266, 12.18991171378460357827408044019
