| 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
−16.82099913489325052840034233588, −15.58437909904220670194407322623, −14.92141027025916791553489566245, −12.57064006206471635212967596156, −11.99582092629293285063692804500, −10.41822429603919993063127303042, −9.384499722190132301334700218570, −7.15453758168964807877459633980, −6.01768941125118873798312387038, −4.80799560255211574720722037051,
0.53527418577244161081033271344, 4.29423770696332839146505108032, 6.10059551255985522312667487171, 7.52532483980612036073614990838, 9.833897176511731067461092588014, 10.76421110515134700499240081572, 11.89713152302668039225427992626, 12.62392876561766800702523629511, 14.01001397215939109189879690575, 16.31300030769590880760600562126
