| L(s) = 1 | + (0.483 + 1.32i)2-s + (3.82 + 0.674i)3-s + (−1.53 + 1.28i)4-s + (−6.81 − 5.71i)5-s + (0.953 + 5.40i)6-s + (2.55 + 4.42i)7-s + (−2.44 − 1.41i)8-s + (5.70 + 2.07i)9-s + (4.30 − 11.8i)10-s + (2.70 − 4.68i)11-s + (−6.72 + 3.88i)12-s + (−12.3 + 2.17i)13-s + (−4.64 + 5.53i)14-s + (−22.1 − 26.4i)15-s + (0.694 − 3.93i)16-s + (24.2 − 8.81i)17-s + ⋯ |
| L(s) = 1 | + (0.241 + 0.664i)2-s + (1.27 + 0.224i)3-s + (−0.383 + 0.321i)4-s + (−1.36 − 1.14i)5-s + (0.158 + 0.901i)6-s + (0.364 + 0.631i)7-s + (−0.306 − 0.176i)8-s + (0.633 + 0.230i)9-s + (0.430 − 1.18i)10-s + (0.245 − 0.425i)11-s + (−0.560 + 0.323i)12-s + (−0.949 + 0.167i)13-s + (−0.331 + 0.395i)14-s + (−1.47 − 1.76i)15-s + (0.0434 − 0.246i)16-s + (1.42 − 0.518i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.25513 + 0.479610i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25513 + 0.479610i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.483 - 1.32i)T \) |
| 19 | \( 1 + (8.05 - 17.2i)T \) |
| good | 3 | \( 1 + (-3.82 - 0.674i)T + (8.45 + 3.07i)T^{2} \) |
| 5 | \( 1 + (6.81 + 5.71i)T + (4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (-2.55 - 4.42i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-2.70 + 4.68i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (12.3 - 2.17i)T + (158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (-24.2 + 8.81i)T + (221. - 185. i)T^{2} \) |
| 23 | \( 1 + (-6.41 + 5.38i)T + (91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (16.1 - 44.4i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-19.0 + 11.0i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 16.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (5.00 + 0.882i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (-28.8 - 24.1i)T + (321. + 1.82e3i)T^{2} \) |
| 47 | \( 1 + (42.1 + 15.3i)T + (1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (18.1 + 21.6i)T + (-487. + 2.76e3i)T^{2} \) |
| 59 | \( 1 + (14.4 + 39.6i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-37.9 + 31.8i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (8.84 - 24.2i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-38.5 + 45.9i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (-5.90 + 33.4i)T + (-5.00e3 - 1.82e3i)T^{2} \) |
| 79 | \( 1 + (-71.9 - 12.6i)T + (5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (35.5 + 61.5i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (51.5 - 9.08i)T + (7.44e3 - 2.70e3i)T^{2} \) |
| 97 | \( 1 + (-14.8 - 40.8i)T + (-7.20e3 + 6.04e3i)T^{2} \) |
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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
−16.07167679896799164547334190078, −14.91517501733230891164239719479, −14.36343763674632161119816844032, −12.70678026726133116071751979045, −11.83941804646940485043789633838, −9.357453709293122321829925764997, −8.393694324750426163939135641323, −7.68055925214490964128593026466, −5.07565587179852025174987326500, −3.57526098277016338901228011002,
2.83885143311570313467525604594, 4.10239903060823673299336638440, 7.26501249148551026499312388586, 8.096329954388783096332752392708, 9.887644521910368982617996561757, 11.14973064115034356420469582770, 12.30280800346220555885485439070, 13.81545539799483001596920445337, 14.73835702492737132288234948254, 15.23294227429759066684812586648
