| L(s) = 1 | + (6.08 + 7.02i)2-s + (−13.1 − 8.42i)3-s + (−3.19 + 22.2i)4-s + (−77.8 − 35.5i)5-s + (−20.6 − 143. i)6-s + (−42.4 + 144. i)7-s + (324. − 208. i)8-s + (100. + 221. i)9-s + (−224. − 763. i)10-s + (1.46e3 + 1.26e3i)11-s + (229. − 264. i)12-s + (2.66e3 − 783. i)13-s + (−1.27e3 + 581. i)14-s + (720. + 1.12e3i)15-s + (4.82e3 + 1.41e3i)16-s + (481. − 69.1i)17-s + ⋯ |
| L(s) = 1 | + (0.761 + 0.878i)2-s + (−0.485 − 0.312i)3-s + (−0.0499 + 0.347i)4-s + (−0.622 − 0.284i)5-s + (−0.0954 − 0.664i)6-s + (−0.123 + 0.421i)7-s + (0.634 − 0.407i)8-s + (0.138 + 0.303i)9-s + (−0.224 − 0.763i)10-s + (1.09 + 0.952i)11-s + (0.132 − 0.153i)12-s + (1.21 − 0.356i)13-s + (−0.464 + 0.211i)14-s + (0.213 + 0.332i)15-s + (1.17 + 0.345i)16-s + (0.0979 − 0.0140i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 - 0.660i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.750 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.29054 + 0.864715i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.29054 + 0.864715i\) |
| \(L(4)\) |
|
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 | \( 1 + (-8.12e3 + 9.05e3i)T \) |
| good | 2 | \( 1 + (-6.08 - 7.02i)T + (-9.10 + 63.3i)T^{2} \) |
| 5 | \( 1 + (77.8 + 35.5i)T + (1.02e4 + 1.18e4i)T^{2} \) |
| 7 | \( 1 + (42.4 - 144. i)T + (-9.89e4 - 6.36e4i)T^{2} \) |
| 11 | \( 1 + (-1.46e3 - 1.26e3i)T + (2.52e5 + 1.75e6i)T^{2} \) |
| 13 | \( 1 + (-2.66e3 + 783. i)T + (4.06e6 - 2.60e6i)T^{2} \) |
| 17 | \( 1 + (-481. + 69.1i)T + (2.31e7 - 6.80e6i)T^{2} \) |
| 19 | \( 1 + (-5.24e3 - 753. i)T + (4.51e7 + 1.32e7i)T^{2} \) |
| 29 | \( 1 + (-3.33e3 - 2.31e4i)T + (-5.70e8 + 1.67e8i)T^{2} \) |
| 31 | \( 1 + (-450. + 289. i)T + (3.68e8 - 8.07e8i)T^{2} \) |
| 37 | \( 1 + (7.06e4 - 3.22e4i)T + (1.68e9 - 1.93e9i)T^{2} \) |
| 41 | \( 1 + (5.19e3 - 1.13e4i)T + (-3.11e9 - 3.58e9i)T^{2} \) |
| 43 | \( 1 + (-1.41e4 + 2.20e4i)T + (-2.62e9 - 5.75e9i)T^{2} \) |
| 47 | \( 1 - 1.24e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + (-5.47e4 + 1.86e5i)T + (-1.86e10 - 1.19e10i)T^{2} \) |
| 59 | \( 1 + (-2.17e5 + 6.38e4i)T + (3.54e10 - 2.28e10i)T^{2} \) |
| 61 | \( 1 + (-1.22e4 - 1.90e4i)T + (-2.14e10 + 4.68e10i)T^{2} \) |
| 67 | \( 1 + (3.87e5 - 3.35e5i)T + (1.28e10 - 8.95e10i)T^{2} \) |
| 71 | \( 1 + (3.73e5 + 4.30e5i)T + (-1.82e10 + 1.26e11i)T^{2} \) |
| 73 | \( 1 + (-2.40e4 + 1.67e5i)T + (-1.45e11 - 4.26e10i)T^{2} \) |
| 79 | \( 1 + (-1.85e5 - 6.31e5i)T + (-2.04e11 + 1.31e11i)T^{2} \) |
| 83 | \( 1 + (1.98e5 - 9.05e4i)T + (2.14e11 - 2.47e11i)T^{2} \) |
| 89 | \( 1 + (-1.91e5 + 2.97e5i)T + (-2.06e11 - 4.52e11i)T^{2} \) |
| 97 | \( 1 + (-8.51e4 - 3.88e4i)T + (5.45e11 + 6.29e11i)T^{2} \) |
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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
−13.74374410566558855942726071338, −12.61205434135120866274592567544, −11.80375960084031496817525385571, −10.37783757336731188636128133622, −8.731228990868883174662255256586, −7.29752768416026016897042193739, −6.35546970042306749106956841858, −5.14908840295566838464451990077, −3.88827133617296151884222667038, −1.18875243479498262997456217036,
1.12676982582477689367810315544, 3.41876884055960041758702870350, 4.05586982803725878509550605937, 5.76196582002345387349079365050, 7.31420186361751805077526687127, 8.931267352879408570524683359477, 10.56593431164937026553788869448, 11.42827676621089623518017310712, 11.90058561554872432011044238387, 13.41274602582870752309744465971
