| L(s) = 1 | + (1.87 − 0.684i)2-s + (−1.31 + 7.43i)3-s + (3.06 − 2.57i)4-s + (3.45 + 2.89i)5-s + (2.62 + 14.8i)6-s + (4.21 + 7.29i)7-s + (4.00 − 6.92i)8-s + (−28.1 − 10.2i)9-s + (8.47 + 3.08i)10-s + (19.9 − 34.5i)11-s + (15.0 + 26.1i)12-s + (−12.5 − 71.0i)13-s + (12.9 + 10.8i)14-s + (−26.0 + 21.8i)15-s + (2.77 − 15.7i)16-s + (−83.5 + 30.4i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (−0.252 + 1.43i)3-s + (0.383 − 0.321i)4-s + (0.309 + 0.259i)5-s + (0.178 + 1.01i)6-s + (0.227 + 0.393i)7-s + (0.176 − 0.306i)8-s + (−1.04 − 0.379i)9-s + (0.268 + 0.0975i)10-s + (0.546 − 0.946i)11-s + (0.363 + 0.628i)12-s + (−0.267 − 1.51i)13-s + (0.246 + 0.206i)14-s + (−0.448 + 0.376i)15-s + (0.0434 − 0.246i)16-s + (−1.19 + 0.433i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.59828 + 0.688981i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59828 + 0.688981i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.87 + 0.684i)T \) |
| 19 | \( 1 + (-34.2 - 75.4i)T \) |
| good | 3 | \( 1 + (1.31 - 7.43i)T + (-25.3 - 9.23i)T^{2} \) |
| 5 | \( 1 + (-3.45 - 2.89i)T + (21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (-4.21 - 7.29i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-19.9 + 34.5i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (12.5 + 71.0i)T + (-2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (83.5 - 30.4i)T + (3.76e3 - 3.15e3i)T^{2} \) |
| 23 | \( 1 + (-68.3 + 57.3i)T + (2.11e3 - 1.19e4i)T^{2} \) |
| 29 | \( 1 + (-163. - 59.5i)T + (1.86e4 + 1.56e4i)T^{2} \) |
| 31 | \( 1 + (-27.1 - 47.0i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 342.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (55.4 - 314. i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 + (155. + 130. i)T + (1.38e4 + 7.82e4i)T^{2} \) |
| 47 | \( 1 + (-91.4 - 33.2i)T + (7.95e4 + 6.67e4i)T^{2} \) |
| 53 | \( 1 + (324. - 272. i)T + (2.58e4 - 1.46e5i)T^{2} \) |
| 59 | \( 1 + (285. - 104. i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (253. - 213. i)T + (3.94e4 - 2.23e5i)T^{2} \) |
| 67 | \( 1 + (878. + 319. i)T + (2.30e5 + 1.93e5i)T^{2} \) |
| 71 | \( 1 + (-818. - 686. i)T + (6.21e4 + 3.52e5i)T^{2} \) |
| 73 | \( 1 + (-179. + 1.02e3i)T + (-3.65e5 - 1.33e5i)T^{2} \) |
| 79 | \( 1 + (-18.3 + 103. i)T + (-4.63e5 - 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-436. - 756. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-27.6 - 156. i)T + (-6.62e5 + 2.41e5i)T^{2} \) |
| 97 | \( 1 + (-926. + 337. i)T + (6.99e5 - 5.86e5i)T^{2} \) |
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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
−15.68620635779161774501590662194, −14.95330722036580180731607582211, −13.81910598042965454214411958509, −12.24712716169790551352139286239, −10.87688312747600208984387670617, −10.20382048080409161737231095238, −8.636275089760946511509392622294, −6.11232305377531277261373522317, −4.87160742175808910652691613972, −3.25787495075168283052231687256,
1.84816249030949792488129624109, 4.72284588952383363859559956597, 6.66114242059269160008647251401, 7.24685101266648355518335980116, 9.158576588949757385797102487390, 11.38875467681758994338043399665, 12.22993613010894853184233776240, 13.42650921670624847010471862522, 14.00643321601545358876467862103, 15.54210829942123554951994193363
