| L(s) = 1 | + (−7.47 − 6.27i)3-s + (−3.19 + 1.16i)5-s + (4.69 + 8.12i)7-s + (11.8 + 67.2i)9-s + (−4.85 + 8.40i)11-s + (−29.5 + 24.8i)13-s + (31.1 + 11.3i)15-s + (−13.5 + 76.6i)17-s + (35.5 − 74.8i)19-s + (15.9 − 90.2i)21-s + (−203. − 74.0i)23-s + (−86.9 + 72.9i)25-s + (201. − 349. i)27-s + (13.3 + 75.5i)29-s + (90.0 + 155. i)31-s + ⋯ |
| L(s) = 1 | + (−1.43 − 1.20i)3-s + (−0.285 + 0.103i)5-s + (0.253 + 0.438i)7-s + (0.439 + 2.49i)9-s + (−0.132 + 0.230i)11-s + (−0.630 + 0.529i)13-s + (0.536 + 0.195i)15-s + (−0.192 + 1.09i)17-s + (0.429 − 0.903i)19-s + (0.165 − 0.937i)21-s + (−1.84 − 0.671i)23-s + (−0.695 + 0.583i)25-s + (1.43 − 2.48i)27-s + (0.0853 + 0.483i)29-s + (0.521 + 0.903i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.236 - 0.971i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.236 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.174894 + 0.222548i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.174894 + 0.222548i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-35.5 + 74.8i)T \) |
| good | 3 | \( 1 + (7.47 + 6.27i)T + (4.68 + 26.5i)T^{2} \) |
| 5 | \( 1 + (3.19 - 1.16i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (-4.69 - 8.12i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (4.85 - 8.40i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (29.5 - 24.8i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (13.5 - 76.6i)T + (-4.61e3 - 1.68e3i)T^{2} \) |
| 23 | \( 1 + (203. + 74.0i)T + (9.32e3 + 7.82e3i)T^{2} \) |
| 29 | \( 1 + (-13.3 - 75.5i)T + (-2.29e4 + 8.34e3i)T^{2} \) |
| 31 | \( 1 + (-90.0 - 155. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 74.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + (195. + 164. i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (425. - 154. i)T + (6.09e4 - 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-64.6 - 366. i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 + (364. + 132. i)T + (1.14e5 + 9.56e4i)T^{2} \) |
| 59 | \( 1 + (12.1 - 68.7i)T + (-1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (-210. - 76.6i)T + (1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (83.9 + 475. i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (933. - 339. i)T + (2.74e5 - 2.30e5i)T^{2} \) |
| 73 | \( 1 + (-705. - 592. i)T + (6.75e4 + 3.83e5i)T^{2} \) |
| 79 | \( 1 + (-551. - 462. i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (558. + 967. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-465. + 390. i)T + (1.22e5 - 6.94e5i)T^{2} \) |
| 97 | \( 1 + (-108. + 615. i)T + (-8.57e5 - 3.12e5i)T^{2} \) |
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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
−14.09048278160861009801520140406, −12.90328399520918061252539716285, −12.05935360686107841009864239588, −11.40699727900288359997699825726, −10.20673792397730641817276707485, −8.238065053318974375705105892883, −7.09658366683761917176150928800, −6.07733185113240908160923250242, −4.80675614068652975217974540453, −1.87182679393775307466413057929,
0.20940353233270688794882610060, 3.87081438290012790543755755410, 5.02441278875655699551444424650, 6.13192958866688516279371208397, 7.81557572705329092587200892847, 9.732361737552463936136713337756, 10.26952536741416447652960860564, 11.61928266468252135134930760332, 12.00544204516350568284640424764, 13.77167450438024217813110212481
