| L(s) = 1 | + (−0.483 − 1.32i)2-s + (−5.23 − 0.922i)3-s + (−1.53 + 1.28i)4-s + (−1.06 − 0.892i)5-s + (1.30 + 7.40i)6-s + (−4.39 − 7.61i)7-s + (2.44 + 1.41i)8-s + (18.0 + 6.58i)9-s + (−0.671 + 1.84i)10-s + (6.05 − 10.4i)11-s + (9.20 − 5.31i)12-s + (−16.6 + 2.92i)13-s + (−7.99 + 9.52i)14-s + (4.74 + 5.65i)15-s + (0.694 − 3.93i)16-s + (14.9 − 5.43i)17-s + ⋯ |
| L(s) = 1 | + (−0.241 − 0.664i)2-s + (−1.74 − 0.307i)3-s + (−0.383 + 0.321i)4-s + (−0.212 − 0.178i)5-s + (0.217 + 1.23i)6-s + (−0.628 − 1.08i)7-s + (0.306 + 0.176i)8-s + (2.00 + 0.731i)9-s + (−0.0671 + 0.184i)10-s + (0.550 − 0.952i)11-s + (0.767 − 0.442i)12-s + (−1.27 + 0.225i)13-s + (−0.571 + 0.680i)14-s + (0.316 + 0.376i)15-s + (0.0434 − 0.246i)16-s + (0.878 − 0.319i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0395111 - 0.347329i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0395111 - 0.347329i\) |
| \(L(2)\) |
|
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 | \( 1 + (15.3 - 11.1i)T \) |
| good | 3 | \( 1 + (5.23 + 0.922i)T + (8.45 + 3.07i)T^{2} \) |
| 5 | \( 1 + (1.06 + 0.892i)T + (4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (4.39 + 7.61i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-6.05 + 10.4i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (16.6 - 2.92i)T + (158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (-14.9 + 5.43i)T + (221. - 185. i)T^{2} \) |
| 23 | \( 1 + (5.07 - 4.25i)T + (91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (-6.87 + 18.8i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-11.7 + 6.81i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 42.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-24.5 - 4.33i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (-30.1 - 25.2i)T + (321. + 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-17.3 - 6.30i)T + (1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (-13.2 - 15.8i)T + (-487. + 2.76e3i)T^{2} \) |
| 59 | \( 1 + (33.7 + 92.7i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (72.1 - 60.5i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (10.8 - 29.8i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (39.8 - 47.5i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (-14.4 + 81.8i)T + (-5.00e3 - 1.82e3i)T^{2} \) |
| 79 | \( 1 + (-42.4 - 7.48i)T + (5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-10.9 - 18.9i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (3.51 - 0.619i)T + (7.44e3 - 2.70e3i)T^{2} \) |
| 97 | \( 1 + (-18.6 - 51.2i)T + (-7.20e3 + 6.04e3i)T^{2} \) |
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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
−16.31300030769590880760600562126, −14.01001397215939109189879690575, −12.62392876561766800702523629511, −11.89713152302668039225427992626, −10.76421110515134700499240081572, −9.833897176511731067461092588014, −7.52532483980612036073614990838, −6.10059551255985522312667487171, −4.29423770696332839146505108032, −0.53527418577244161081033271344,
4.80799560255211574720722037051, 6.01768941125118873798312387038, 7.15453758168964807877459633980, 9.384499722190132301334700218570, 10.41822429603919993063127303042, 11.99582092629293285063692804500, 12.57064006206471635212967596156, 14.92141027025916791553489566245, 15.58437909904220670194407322623, 16.82099913489325052840034233588
