| L(s) = 1 | + (6.34 − 7.32i)2-s + (13.1 − 8.42i)3-s + (−4.25 − 29.6i)4-s + (79.9 − 36.5i)5-s + (21.4 − 149. i)6-s + (−34.3 − 116. i)7-s + (277. + 178. i)8-s + (100. − 221. i)9-s + (239. − 817. i)10-s + (1.33e3 − 1.15e3i)11-s + (−305. − 352. i)12-s + (−2.55e3 − 749. i)13-s + (−1.07e3 − 490. i)14-s + (740. − 1.15e3i)15-s + (4.90e3 − 1.44e3i)16-s + (2.06e3 + 297. i)17-s + ⋯ |
| L(s) = 1 | + (0.793 − 0.915i)2-s + (0.485 − 0.312i)3-s + (−0.0665 − 0.462i)4-s + (0.639 − 0.292i)5-s + (0.0995 − 0.692i)6-s + (−0.100 − 0.340i)7-s + (0.542 + 0.348i)8-s + (0.138 − 0.303i)9-s + (0.239 − 0.817i)10-s + (1.00 − 0.866i)11-s + (−0.176 − 0.203i)12-s + (−1.16 − 0.341i)13-s + (−0.391 − 0.178i)14-s + (0.219 − 0.341i)15-s + (1.19 − 0.351i)16-s + (0.420 + 0.0604i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.238 + 0.971i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.31400 - 2.95047i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.31400 - 2.95047i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-13.1 + 8.42i)T \) |
| 23 | \( 1 + (-4.26e3 + 1.13e4i)T \) |
| good | 2 | \( 1 + (-6.34 + 7.32i)T + (-9.10 - 63.3i)T^{2} \) |
| 5 | \( 1 + (-79.9 + 36.5i)T + (1.02e4 - 1.18e4i)T^{2} \) |
| 7 | \( 1 + (34.3 + 116. i)T + (-9.89e4 + 6.36e4i)T^{2} \) |
| 11 | \( 1 + (-1.33e3 + 1.15e3i)T + (2.52e5 - 1.75e6i)T^{2} \) |
| 13 | \( 1 + (2.55e3 + 749. i)T + (4.06e6 + 2.60e6i)T^{2} \) |
| 17 | \( 1 + (-2.06e3 - 297. i)T + (2.31e7 + 6.80e6i)T^{2} \) |
| 19 | \( 1 + (2.20e3 - 316. i)T + (4.51e7 - 1.32e7i)T^{2} \) |
| 29 | \( 1 + (5.68e3 - 3.95e4i)T + (-5.70e8 - 1.67e8i)T^{2} \) |
| 31 | \( 1 + (2.07e4 + 1.33e4i)T + (3.68e8 + 8.07e8i)T^{2} \) |
| 37 | \( 1 + (-8.54e3 - 3.90e3i)T + (1.68e9 + 1.93e9i)T^{2} \) |
| 41 | \( 1 + (7.15e3 + 1.56e4i)T + (-3.11e9 + 3.58e9i)T^{2} \) |
| 43 | \( 1 + (-3.29e4 - 5.12e4i)T + (-2.62e9 + 5.75e9i)T^{2} \) |
| 47 | \( 1 + 2.70e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + (-6.99e4 - 2.38e5i)T + (-1.86e10 + 1.19e10i)T^{2} \) |
| 59 | \( 1 + (-2.47e5 - 7.25e4i)T + (3.54e10 + 2.28e10i)T^{2} \) |
| 61 | \( 1 + (-8.13e3 + 1.26e4i)T + (-2.14e10 - 4.68e10i)T^{2} \) |
| 67 | \( 1 + (-2.40e5 - 2.08e5i)T + (1.28e10 + 8.95e10i)T^{2} \) |
| 71 | \( 1 + (3.85e4 - 4.44e4i)T + (-1.82e10 - 1.26e11i)T^{2} \) |
| 73 | \( 1 + (-7.07e4 - 4.91e5i)T + (-1.45e11 + 4.26e10i)T^{2} \) |
| 79 | \( 1 + (2.44e5 - 8.32e5i)T + (-2.04e11 - 1.31e11i)T^{2} \) |
| 83 | \( 1 + (6.60e5 + 3.01e5i)T + (2.14e11 + 2.47e11i)T^{2} \) |
| 89 | \( 1 + (3.01e5 + 4.69e5i)T + (-2.06e11 + 4.52e11i)T^{2} \) |
| 97 | \( 1 + (-2.34e5 + 1.07e5i)T + (5.45e11 - 6.29e11i)T^{2} \) |
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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
−13.04861963783614702229351684816, −12.38769714017787684188443990159, −11.17565400646555083789956414857, −9.960073376920508053443229077043, −8.690178466303570703154170786272, −7.19073852120360397835738090867, −5.49636645328606479670481351048, −3.96634524821112404854562797025, −2.67292546260540488801637713886, −1.24529470755888444517269850472,
2.05613057775508543495141460604, 4.00744456041198647627691453905, 5.27236642178055272922102820496, 6.54448235302875323532640987113, 7.57925849412522754775823364878, 9.375343715548884018235211384029, 10.09674980483536965028123210138, 11.91222365929228365759957004025, 13.14193724287351169530127586712, 14.21107942772873620833564082653
