| L(s) = 1 | + (−1.23 + 3.80i)2-s − 20.2·3-s + (−12.9 − 9.40i)4-s + (−77.2 − 56.1i)5-s + (25.0 − 76.9i)6-s + (−23.9 − 73.8i)7-s + (51.7 − 37.6i)8-s + 166.·9-s + (309. − 224. i)10-s + (−303. + 220. i)11-s + (261. + 190. i)12-s + (63.8 − 196. i)13-s + 310.·14-s + (1.56e3 + 1.13e3i)15-s + (79.1 + 243. i)16-s + (−1.34e3 + 974. i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s − 1.29·3-s + (−0.404 − 0.293i)4-s + (−1.38 − 1.00i)5-s + (0.283 − 0.872i)6-s + (−0.185 − 0.569i)7-s + (0.286 − 0.207i)8-s + 0.683·9-s + (0.977 − 0.710i)10-s + (−0.755 + 0.549i)11-s + (0.524 + 0.381i)12-s + (0.104 − 0.322i)13-s + 0.423·14-s + (1.79 + 1.30i)15-s + (0.0772 + 0.237i)16-s + (−1.12 + 0.818i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 - 0.955i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.183875 + 0.135469i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.183875 + 0.135469i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 - 3.80i)T \) |
| 41 | \( 1 + (-9.35e3 - 5.31e3i)T \) |
| good | 3 | \( 1 + 20.2T + 243T^{2} \) |
| 5 | \( 1 + (77.2 + 56.1i)T + (965. + 2.97e3i)T^{2} \) |
| 7 | \( 1 + (23.9 + 73.8i)T + (-1.35e4 + 9.87e3i)T^{2} \) |
| 11 | \( 1 + (303. - 220. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-63.8 + 196. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (1.34e3 - 974. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (562. + 1.73e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (34.4 - 106. i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-1.09e3 - 795. i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-1.33e3 + 968. i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (2.08e3 + 1.51e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 43 | \( 1 + (3.46e3 - 1.06e4i)T + (-1.18e8 - 8.64e7i)T^{2} \) |
| 47 | \( 1 + (-7.16e3 + 2.20e4i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (5.20e3 + 3.77e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (6.39e3 - 1.96e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (1.00e4 + 3.09e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (3.76e4 + 2.73e4i)T + (4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-2.12e4 + 1.54e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 - 4.80e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.03e5T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.23e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-4.37e4 - 1.34e5i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-9.02e4 - 6.55e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−13.23991283230838442932686543334, −12.52035904425700383922014645517, −11.35586885122955787140786225338, −10.49182421280199092261315351381, −8.812145288674324976911061443438, −7.72364084170319204140912325908, −6.59216521992971342680691844042, −5.08580004839465871188356594747, −4.27967166206187461932890194333, −0.61591740383979648813092161546,
0.23567752400195752672631818770, 2.83950336040576945614656152341, 4.37431098561085197005310315244, 5.97964281446657437171058904175, 7.28437855514492982883860335659, 8.610585948982355872132999327664, 10.38820239816964987506821604532, 11.12172251292983985481702367225, 11.74271769709764186097076657431, 12.57143633475963816981135889672
