| L(s) = 1 | + (4.10 + 3.27i)2-s + (−3.10 − 20.5i)3-s + (−8.10 − 35.5i)4-s + (95.1 + 139. i)5-s + (54.6 − 94.7i)6-s + (−111. + 64.3i)7-s + (228. − 475. i)8-s + (282. − 87.0i)9-s + (−66.3 + 884. i)10-s + (502. − 2.20e3i)11-s + (−705. + 277. i)12-s + (−305. − 4.07e3i)13-s + (−668. − 100. i)14-s + (2.57e3 − 2.39e3i)15-s + (395. − 190. i)16-s + (4.03e3 + 2.75e3i)17-s + ⋯ |
| L(s) = 1 | + (0.513 + 0.409i)2-s + (−0.114 − 0.762i)3-s + (−0.126 − 0.554i)4-s + (0.761 + 1.11i)5-s + (0.253 − 0.438i)6-s + (−0.324 + 0.187i)7-s + (0.446 − 0.928i)8-s + (0.387 − 0.119i)9-s + (−0.0663 + 0.884i)10-s + (0.377 − 1.65i)11-s + (−0.408 + 0.160i)12-s + (−0.139 − 1.85i)13-s + (−0.243 − 0.0366i)14-s + (0.764 − 0.709i)15-s + (0.0966 − 0.0465i)16-s + (0.820 + 0.559i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.12585 - 0.998324i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.12585 - 0.998324i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (3.94e4 - 6.90e4i)T \) |
| good | 2 | \( 1 + (-4.10 - 3.27i)T + (14.2 + 62.3i)T^{2} \) |
| 3 | \( 1 + (3.10 + 20.5i)T + (-696. + 214. i)T^{2} \) |
| 5 | \( 1 + (-95.1 - 139. i)T + (-5.70e3 + 1.45e4i)T^{2} \) |
| 7 | \( 1 + (111. - 64.3i)T + (5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-502. + 2.20e3i)T + (-1.59e6 - 7.68e5i)T^{2} \) |
| 13 | \( 1 + (305. + 4.07e3i)T + (-4.77e6 + 7.19e5i)T^{2} \) |
| 17 | \( 1 + (-4.03e3 - 2.75e3i)T + (8.81e6 + 2.24e7i)T^{2} \) |
| 19 | \( 1 + (2.86e3 - 9.29e3i)T + (-3.88e7 - 2.65e7i)T^{2} \) |
| 23 | \( 1 + (-2.09e3 - 1.94e3i)T + (1.10e7 + 1.47e8i)T^{2} \) |
| 29 | \( 1 + (779. - 5.17e3i)T + (-5.68e8 - 1.75e8i)T^{2} \) |
| 31 | \( 1 + (-1.86e3 - 4.74e3i)T + (-6.50e8 + 6.03e8i)T^{2} \) |
| 37 | \( 1 + (-8.16e4 - 4.71e4i)T + (1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-2.49e4 + 3.13e4i)T + (-1.05e9 - 4.63e9i)T^{2} \) |
| 47 | \( 1 + (1.95e4 + 8.58e4i)T + (-9.71e9 + 4.67e9i)T^{2} \) |
| 53 | \( 1 + (2.16e4 - 2.89e5i)T + (-2.19e10 - 3.30e9i)T^{2} \) |
| 59 | \( 1 + (-1.18e5 + 5.68e4i)T + (2.62e10 - 3.29e10i)T^{2} \) |
| 61 | \( 1 + (1.90e5 + 7.47e4i)T + (3.77e10 + 3.50e10i)T^{2} \) |
| 67 | \( 1 + (4.92e5 + 1.51e5i)T + (7.47e10 + 5.09e10i)T^{2} \) |
| 71 | \( 1 + (4.02e4 + 4.34e4i)T + (-9.57e9 + 1.27e11i)T^{2} \) |
| 73 | \( 1 + (-5.22e5 + 3.91e4i)T + (1.49e11 - 2.25e10i)T^{2} \) |
| 79 | \( 1 + (-1.78e5 - 3.09e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (3.19e5 - 4.81e4i)T + (3.12e11 - 9.63e10i)T^{2} \) |
| 89 | \( 1 + (-1.83e4 - 1.21e5i)T + (-4.74e11 + 1.46e11i)T^{2} \) |
| 97 | \( 1 + (-5.10e4 + 2.23e5i)T + (-7.50e11 - 3.61e11i)T^{2} \) |
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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
−14.43198180088215334113575542411, −13.51567944815524106502235869622, −12.59902307523575045205501266301, −10.71888479976797440363960348845, −9.923025646294065864469720412593, −7.83170001835021784977461754083, −6.19302095719974858410988566446, −5.92398562441513824853898905594, −3.25667567891520513398512737140, −1.08496323349528420425964950882,
1.96019325464620679757089200649, 4.30202391836180621645438027727, 4.83278758376422345242263952431, 7.12046716554849867966765068303, 9.149696766008588840822701651817, 9.737936125759432623914954151138, 11.52371388576803592292806973471, 12.62985303597943336452524370228, 13.39945892271165259835473808406, 14.69276855009703769651214543131
