| L(s) = 1 | + (−1.14 + 0.172i)2-s + (1.61 + 1.10i)3-s + (−0.634 + 0.195i)4-s + (0.136 − 1.82i)5-s + (−2.04 − 0.983i)6-s + (−1.31 + 2.29i)7-s + (2.77 − 1.33i)8-s + (0.308 + 0.785i)9-s + (0.158 + 2.11i)10-s + (1.56 − 3.98i)11-s + (−1.24 − 0.383i)12-s + (−0.623 − 0.781i)13-s + (1.10 − 2.85i)14-s + (2.23 − 2.80i)15-s + (−1.84 + 1.25i)16-s + (5.56 − 5.16i)17-s + ⋯ |
| L(s) = 1 | + (−0.808 + 0.121i)2-s + (0.935 + 0.637i)3-s + (−0.317 + 0.0977i)4-s + (0.0612 − 0.816i)5-s + (−0.833 − 0.401i)6-s + (−0.495 + 0.868i)7-s + (0.980 − 0.472i)8-s + (0.102 + 0.261i)9-s + (0.0500 + 0.667i)10-s + (0.471 − 1.20i)11-s + (−0.358 − 0.110i)12-s + (−0.172 − 0.216i)13-s + (0.294 − 0.762i)14-s + (0.578 − 0.724i)15-s + (−0.461 + 0.314i)16-s + (1.35 − 1.25i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0110i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.17912 - 0.00653716i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17912 - 0.00653716i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (1.31 - 2.29i)T \) |
| 13 | \( 1 + (0.623 + 0.781i)T \) |
| good | 2 | \( 1 + (1.14 - 0.172i)T + (1.91 - 0.589i)T^{2} \) |
| 3 | \( 1 + (-1.61 - 1.10i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (-0.136 + 1.82i)T + (-4.94 - 0.745i)T^{2} \) |
| 11 | \( 1 + (-1.56 + 3.98i)T + (-8.06 - 7.48i)T^{2} \) |
| 17 | \( 1 + (-5.56 + 5.16i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (2.12 - 3.67i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.53 - 6.06i)T + (1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (1.32 + 5.78i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (0.638 + 1.10i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.83 - 0.565i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (0.00976 - 0.00470i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-7.66 - 3.69i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (2.99 - 0.451i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (1.96 - 0.605i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.564 - 7.52i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-7.53 - 2.32i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (5.07 + 8.78i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.556 - 2.43i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-8.96 - 1.35i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (2.73 - 4.73i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.68 + 10.8i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (2.71 + 6.91i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + 5.12T + 97T^{2} \) |
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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
−9.980707763836953029413976024594, −9.413042850827014162321079066393, −9.002556685515322551871809479301, −8.323243384113286617506170678862, −7.53707629146688132702037931549, −5.97262845696965069221234037498, −5.03724390341343167322720511539, −3.77155217143211712963038507480, −2.94991050939562437275609222039, −0.943182636852975303617557489611,
1.28294900096729148886889658849, 2.51726868755544275321913716587, 3.74054889045077870959442428022, 4.91673497785134999676375546736, 6.72300414495947991392546216162, 7.14719141322516957807117795303, 7.994702767825036938240472544257, 8.882381205076407503345776361551, 9.615458421669927832187077232078, 10.53857048205828803857580874245
