| L(s) = 1 | + (−0.856 − 2.63i)2-s + (0.254 − 0.184i)4-s + (−7.75 + 8.04i)5-s − 7.15·7-s + (−18.6 − 13.5i)8-s + (27.8 + 13.5i)10-s + (15.3 + 47.3i)11-s + (16.6 − 51.1i)13-s + (6.13 + 18.8i)14-s + (−18.9 + 58.3i)16-s + (−10.7 − 7.80i)17-s + (124. + 90.3i)19-s + (−0.486 + 3.47i)20-s + (111. − 81.1i)22-s + (66.4 + 204. i)23-s + ⋯ |
| L(s) = 1 | + (−0.302 − 0.932i)2-s + (0.0317 − 0.0230i)4-s + (−0.694 + 0.719i)5-s − 0.386·7-s + (−0.824 − 0.598i)8-s + (0.881 + 0.428i)10-s + (0.421 + 1.29i)11-s + (0.354 − 1.09i)13-s + (0.117 + 0.360i)14-s + (−0.296 + 0.912i)16-s + (−0.153 − 0.111i)17-s + (1.50 + 1.09i)19-s + (−0.00543 + 0.0389i)20-s + (1.08 − 0.786i)22-s + (0.602 + 1.85i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0992i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.17713 + 0.0585579i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17713 + 0.0585579i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (7.75 - 8.04i)T \) |
| good | 2 | \( 1 + (0.856 + 2.63i)T + (-6.47 + 4.70i)T^{2} \) |
| 7 | \( 1 + 7.15T + 343T^{2} \) |
| 11 | \( 1 + (-15.3 - 47.3i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (-16.6 + 51.1i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (10.7 + 7.80i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-124. - 90.3i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-66.4 - 204. i)T + (-9.84e3 + 7.15e3i)T^{2} \) |
| 29 | \( 1 + (56.2 - 40.8i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-122. - 89.1i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (92.0 - 283. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-110. + 341. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 - 451.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-124. + 90.6i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-39.5 + 28.7i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (9.36 - 28.8i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-175. - 539. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (557. + 404. i)T + (9.29e4 + 2.86e5i)T^{2} \) |
| 71 | \( 1 + (684. - 497. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (110. + 339. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (158. - 115. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-580. - 421. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + (-371. - 1.14e3i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (1.00e3 - 727. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−11.86816421747461517645114489528, −10.79400923446614222789351353162, −10.05818509289369848487398654792, −9.293782837104292592268815902201, −7.71521058269666745752764026751, −6.93316800358164931348206113392, −5.60403648488081522809283611546, −3.75531277693555148599301484930, −2.92770545213649202831126764494, −1.27961670617556655951476233217,
0.60785052503721146309770686582, 3.00569272612202007273863800252, 4.43847740781161492768045930481, 5.85170546628543704688021792926, 6.76987726159489633621080704093, 7.76867584084116337367832562621, 8.827768197488056863878396612278, 9.199533708634069953638510545179, 11.14572102208062200562138196697, 11.63143333596692393635139378274
