| L(s) = 1 | + (0.830 + 1.81i)2-s + (2.87 − 0.845i)3-s + (−2.61 + 3.02i)4-s + (−8.49 − 5.45i)5-s + (3.92 + 4.53i)6-s + (4.18 + 29.0i)7-s + (−7.67 − 2.25i)8-s + (7.57 − 4.86i)9-s + (2.87 − 19.9i)10-s + (−8.00 + 17.5i)11-s + (−4.98 + 10.9i)12-s + (−12.3 + 85.9i)13-s + (−49.4 + 31.7i)14-s + (−29.0 − 8.53i)15-s + (−2.27 − 15.8i)16-s + (31.7 + 36.6i)17-s + ⋯ |
| L(s) = 1 | + (0.293 + 0.643i)2-s + (0.553 − 0.162i)3-s + (−0.327 + 0.377i)4-s + (−0.759 − 0.488i)5-s + (0.267 + 0.308i)6-s + (0.225 + 1.57i)7-s + (−0.339 − 0.0996i)8-s + (0.280 − 0.180i)9-s + (0.0908 − 0.632i)10-s + (−0.219 + 0.480i)11-s + (−0.119 + 0.262i)12-s + (−0.263 + 1.83i)13-s + (−0.943 + 0.606i)14-s + (−0.500 − 0.146i)15-s + (−0.0355 − 0.247i)16-s + (0.453 + 0.523i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.545 - 0.838i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.807383 + 1.48890i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.807383 + 1.48890i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.830 - 1.81i)T \) |
| 3 | \( 1 + (-2.87 + 0.845i)T \) |
| 23 | \( 1 + (-81.0 + 74.7i)T \) |
| good | 5 | \( 1 + (8.49 + 5.45i)T + (51.9 + 113. i)T^{2} \) |
| 7 | \( 1 + (-4.18 - 29.0i)T + (-329. + 96.6i)T^{2} \) |
| 11 | \( 1 + (8.00 - 17.5i)T + (-871. - 1.00e3i)T^{2} \) |
| 13 | \( 1 + (12.3 - 85.9i)T + (-2.10e3 - 618. i)T^{2} \) |
| 17 | \( 1 + (-31.7 - 36.6i)T + (-699. + 4.86e3i)T^{2} \) |
| 19 | \( 1 + (-5.59 + 6.46i)T + (-976. - 6.78e3i)T^{2} \) |
| 29 | \( 1 + (-39.7 - 45.8i)T + (-3.47e3 + 2.41e4i)T^{2} \) |
| 31 | \( 1 + (296. + 87.1i)T + (2.50e4 + 1.61e4i)T^{2} \) |
| 37 | \( 1 + (16.3 - 10.5i)T + (2.10e4 - 4.60e4i)T^{2} \) |
| 41 | \( 1 + (-297. - 191. i)T + (2.86e4 + 6.26e4i)T^{2} \) |
| 43 | \( 1 + (-302. + 88.8i)T + (6.68e4 - 4.29e4i)T^{2} \) |
| 47 | \( 1 + 104.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (77.5 + 539. i)T + (-1.42e5 + 4.19e4i)T^{2} \) |
| 59 | \( 1 + (-39.7 + 276. i)T + (-1.97e5 - 5.78e4i)T^{2} \) |
| 61 | \( 1 + (-396. - 116. i)T + (1.90e5 + 1.22e5i)T^{2} \) |
| 67 | \( 1 + (-336. - 736. i)T + (-1.96e5 + 2.27e5i)T^{2} \) |
| 71 | \( 1 + (-262. - 575. i)T + (-2.34e5 + 2.70e5i)T^{2} \) |
| 73 | \( 1 + (-403. + 465. i)T + (-5.53e4 - 3.85e5i)T^{2} \) |
| 79 | \( 1 + (32.2 - 224. i)T + (-4.73e5 - 1.38e5i)T^{2} \) |
| 83 | \( 1 + (-1.11e3 + 716. i)T + (2.37e5 - 5.20e5i)T^{2} \) |
| 89 | \( 1 + (-263. + 77.2i)T + (5.93e5 - 3.81e5i)T^{2} \) |
| 97 | \( 1 + (-910. - 584. i)T + (3.79e5 + 8.30e5i)T^{2} \) |
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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
−12.90666145913980220236185672723, −12.28980177318174860347918766120, −11.42632592418185559057238933044, −9.402818671475880753965797231919, −8.768896510985572612811932738368, −7.82791797365536385054625414521, −6.62713962577627960268597383109, −5.19703954490835054337038271769, −4.05214373322671541619434374527, −2.23497217700796603635258358709,
0.74637282503352223504454679055, 3.08652587969805504623701343321, 3.84519127549569794364068303348, 5.31578796634634289851251036353, 7.35807275841678955121121052149, 7.87222016563876813424652724695, 9.487918602790221892096121663610, 10.73437163603435147141471487125, 10.92262107973500750167591194539, 12.48512186597825710962432522225
