| L(s) = 1 | + (−3.75 + 1.36i)2-s + (3.93 − 22.3i)3-s + (12.2 − 10.2i)4-s + (61.0 + 51.2i)5-s + (15.7 + 89.2i)6-s + (−95.8 − 166. i)7-s + (−32.0 + 55.4i)8-s + (−253. − 92.3i)9-s + (−299. − 109. i)10-s + (179. − 310. i)11-s + (−181. − 313. i)12-s + (−77.0 − 437. i)13-s + (587. + 493. i)14-s + (1.38e3 − 1.15e3i)15-s + (44.4 − 252. i)16-s + (−931. + 339. i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 + 0.241i)2-s + (0.252 − 1.43i)3-s + (0.383 − 0.321i)4-s + (1.09 + 0.916i)5-s + (0.178 + 1.01i)6-s + (−0.739 − 1.28i)7-s + (−0.176 + 0.306i)8-s + (−1.04 − 0.379i)9-s + (−0.947 − 0.344i)10-s + (0.446 − 0.773i)11-s + (−0.363 − 0.629i)12-s + (−0.126 − 0.717i)13-s + (0.801 + 0.672i)14-s + (1.58 − 1.33i)15-s + (0.0434 − 0.246i)16-s + (−0.782 + 0.284i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.809751 - 0.984818i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.809751 - 0.984818i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.75 - 1.36i)T \) |
| 19 | \( 1 + (1.31e3 + 865. i)T \) |
| good | 3 | \( 1 + (-3.93 + 22.3i)T + (-228. - 83.1i)T^{2} \) |
| 5 | \( 1 + (-61.0 - 51.2i)T + (542. + 3.07e3i)T^{2} \) |
| 7 | \( 1 + (95.8 + 166. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-179. + 310. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (77.0 + 437. i)T + (-3.48e5 + 1.26e5i)T^{2} \) |
| 17 | \( 1 + (931. - 339. i)T + (1.08e6 - 9.12e5i)T^{2} \) |
| 23 | \( 1 + (-1.02e3 + 860. i)T + (1.11e6 - 6.33e6i)T^{2} \) |
| 29 | \( 1 + (-4.76e3 - 1.73e3i)T + (1.57e7 + 1.31e7i)T^{2} \) |
| 31 | \( 1 + (-3.27e3 - 5.67e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 - 1.32e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (1.71e3 - 9.73e3i)T + (-1.08e8 - 3.96e7i)T^{2} \) |
| 43 | \( 1 + (-8.63e3 - 7.24e3i)T + (2.55e7 + 1.44e8i)T^{2} \) |
| 47 | \( 1 + (2.64e4 + 9.63e3i)T + (1.75e8 + 1.47e8i)T^{2} \) |
| 53 | \( 1 + (-2.34e4 + 1.96e4i)T + (7.26e7 - 4.11e8i)T^{2} \) |
| 59 | \( 1 + (-3.86e4 + 1.40e4i)T + (5.47e8 - 4.59e8i)T^{2} \) |
| 61 | \( 1 + (1.22e4 - 1.03e4i)T + (1.46e8 - 8.31e8i)T^{2} \) |
| 67 | \( 1 + (2.08e4 + 7.58e3i)T + (1.03e9 + 8.67e8i)T^{2} \) |
| 71 | \( 1 + (-1.27e4 - 1.06e4i)T + (3.13e8 + 1.77e9i)T^{2} \) |
| 73 | \( 1 + (-1.05e3 + 5.99e3i)T + (-1.94e9 - 7.09e8i)T^{2} \) |
| 79 | \( 1 + (1.57e4 - 8.91e4i)T + (-2.89e9 - 1.05e9i)T^{2} \) |
| 83 | \( 1 + (-5.04e4 - 8.74e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-1.90e4 - 1.08e5i)T + (-5.24e9 + 1.90e9i)T^{2} \) |
| 97 | \( 1 + (-7.68e4 + 2.79e4i)T + (6.57e9 - 5.51e9i)T^{2} \) |
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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
−14.65723505712599669481724664208, −13.64287419401907015723389046090, −12.98215311480584290914374189021, −10.98688148519810062901526311954, −9.986902914081763272786270746507, −8.317309188933994722623358717142, −6.72835164309404031987688224390, −6.51462290298638832705527202682, −2.70041872433812858539677759927, −0.899697631745169269183301659300,
2.30746764115570162124907178238, 4.50973574501255352369787023659, 6.12228321988978055630866135651, 8.869741744795874366078960275228, 9.326550032571696690196975615674, 10.11450406191731310494818729719, 11.86060677223005348180810799733, 13.09861668224859004865362470390, 14.82399017856863457269995485553, 15.81196485887924461829078311839
