| L(s) = 1 | + (−1.60 + 35.7i)2-s + (−201. − 175. i)3-s + (−765. − 68.8i)4-s + (−311. − 957. i)5-s + (6.60e3 − 6.90e3i)6-s + (3.84e3 + 1.06e3i)7-s + (1.23e3 − 9.08e3i)8-s + (6.92e3 + 5.11e4i)9-s + (3.47e4 − 9.58e3i)10-s + (4.22e3 − 7.85e3i)11-s + (1.41e5 + 1.48e5i)12-s + (−4.15e4 − 7.72e4i)13-s + (−4.40e4 + 1.35e5i)14-s + (−1.05e5 + 2.47e5i)15-s + (−6.43e4 − 1.16e4i)16-s + (3.76e5 + 2.73e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.0709 + 1.57i)2-s + (−1.43 − 1.25i)3-s + (−1.49 − 0.134i)4-s + (−0.222 − 0.685i)5-s + (2.07 − 2.17i)6-s + (0.605 + 0.166i)7-s + (0.106 − 0.783i)8-s + (0.351 + 2.59i)9-s + (1.09 − 0.303i)10-s + (0.0870 − 0.161i)11-s + (1.97 + 2.06i)12-s + (−0.403 − 0.750i)13-s + (−0.306 + 0.943i)14-s + (−0.538 + 1.26i)15-s + (−0.245 − 0.0445i)16-s + (1.09 + 0.794i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.369333 - 0.281960i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.369333 - 0.281960i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + (-2.10e8 - 4.09e7i)T \) |
| good | 2 | \( 1 + (1.60 - 35.7i)T + (-509. - 45.8i)T^{2} \) |
| 3 | \( 1 + (201. + 175. i)T + (2.64e3 + 1.95e4i)T^{2} \) |
| 5 | \( 1 + (311. + 957. i)T + (-1.58e6 + 1.14e6i)T^{2} \) |
| 7 | \( 1 + (-3.84e3 - 1.06e3i)T + (3.46e7 + 2.06e7i)T^{2} \) |
| 11 | \( 1 + (-4.22e3 + 7.85e3i)T + (-1.29e9 - 1.96e9i)T^{2} \) |
| 13 | \( 1 + (4.15e4 + 7.72e4i)T + (-5.84e9 + 8.85e9i)T^{2} \) |
| 17 | \( 1 + (-3.76e5 - 2.73e5i)T + (3.66e10 + 1.12e11i)T^{2} \) |
| 19 | \( 1 + (-3.35e5 - 7.84e5i)T + (-2.22e11 + 2.33e11i)T^{2} \) |
| 23 | \( 1 + (8.25e5 + 1.03e6i)T + (-4.00e11 + 1.75e12i)T^{2} \) |
| 29 | \( 1 + (1.53e6 - 9.19e5i)T + (6.87e12 - 1.27e13i)T^{2} \) |
| 31 | \( 1 + (-6.30e6 + 1.14e6i)T + (2.47e13 - 9.29e12i)T^{2} \) |
| 37 | \( 1 + (6.11e6 - 7.66e6i)T + (-2.89e13 - 1.26e14i)T^{2} \) |
| 41 | \( 1 + (1.85e6 + 8.92e5i)T + (2.04e14 + 2.55e14i)T^{2} \) |
| 43 | \( 1 + (1.21e7 + 4.57e6i)T + (3.78e14 + 3.30e14i)T^{2} \) |
| 47 | \( 1 + (-1.54e6 + 1.35e6i)T + (1.50e14 - 1.10e15i)T^{2} \) |
| 53 | \( 1 + (2.14e7 - 1.93e6i)T + (3.24e15 - 5.89e14i)T^{2} \) |
| 59 | \( 1 + (5.35e7 + 5.60e7i)T + (-3.88e14 + 8.65e15i)T^{2} \) |
| 61 | \( 1 + (-6.49e7 + 1.79e7i)T + (1.00e16 - 5.99e15i)T^{2} \) |
| 67 | \( 1 + (3.01e8 + 2.71e7i)T + (2.67e16 + 4.85e15i)T^{2} \) |
| 73 | \( 1 + (-1.91e7 + 4.26e8i)T + (-5.86e16 - 5.27e15i)T^{2} \) |
| 79 | \( 1 + (-3.80e7 + 2.81e8i)T + (-1.15e17 - 3.18e16i)T^{2} \) |
| 83 | \( 1 + (3.32e7 + 3.47e7i)T + (-8.38e15 + 1.86e17i)T^{2} \) |
| 89 | \( 1 + (-4.56e8 + 4.10e7i)T + (3.44e17 - 6.25e16i)T^{2} \) |
| 97 | \( 1 + (1.25e9 - 6.05e8i)T + (4.73e17 - 5.94e17i)T^{2} \) |
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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
−12.48392465578732357970632519413, −11.90839269866797890473717137240, −10.35433614660692629326410374872, −8.113744384303304429184683655623, −7.900093358949638782921246701621, −6.48020451986647201225255576141, −5.62628663025449580473365822835, −4.86060703784420208211015454028, −1.44681571967336983108494994429, −0.21653471571819276206542208467,
1.07334818167988719908555045730, 3.06725057462135220349085701590, 4.25928517753949715410402144771, 5.19075266183859909986078128609, 6.95730454957850436406309838961, 9.363916174178259369717753689792, 10.00515130031337526273484483646, 11.05168431093371408908212250183, 11.52638959714595455866611140715, 12.20093048867409655715089945965
