L(s) = 1 | + (0.665 − 1.24i)2-s + (1.31 + 1.12i)3-s + (−1.11 − 1.66i)4-s + (2.18 − 0.455i)5-s + (2.28 − 0.892i)6-s − 0.246·7-s + (−2.81 + 0.285i)8-s + (0.463 + 2.96i)9-s + (0.888 − 3.03i)10-s + (−1.26 − 3.90i)11-s + (0.403 − 3.44i)12-s + (3.99 + 1.29i)13-s + (−0.164 + 0.307i)14-s + (3.39 + 1.86i)15-s + (−1.51 + 3.70i)16-s + (−1.31 − 0.954i)17-s + ⋯ |
L(s) = 1 | + (0.470 − 0.882i)2-s + (0.759 + 0.650i)3-s + (−0.557 − 0.830i)4-s + (0.979 − 0.203i)5-s + (0.931 − 0.364i)6-s − 0.0931·7-s + (−0.994 + 0.100i)8-s + (0.154 + 0.987i)9-s + (0.280 − 0.959i)10-s + (−0.382 − 1.17i)11-s + (0.116 − 0.993i)12-s + (1.10 + 0.359i)13-s + (−0.0438 + 0.0822i)14-s + (0.876 + 0.481i)15-s + (−0.379 + 0.925i)16-s + (−0.318 − 0.231i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.92180 - 0.928907i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.92180 - 0.928907i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.665 + 1.24i)T \) |
| 3 | \( 1 + (-1.31 - 1.12i)T \) |
| 5 | \( 1 + (-2.18 + 0.455i)T \) |
good | 7 | \( 1 + 0.246T + 7T^{2} \) |
| 11 | \( 1 + (1.26 + 3.90i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-3.99 - 1.29i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.31 + 0.954i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (3.07 - 4.23i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (0.275 - 0.0896i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (3.35 + 4.61i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (2.15 - 2.96i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (3.54 + 1.15i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-6.74 - 2.19i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 5.82T + 43T^{2} \) |
| 47 | \( 1 + (1.31 + 1.80i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (6.78 - 4.93i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.91 + 5.87i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-3.54 - 10.9i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-5.77 - 4.19i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (12.8 - 9.35i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-5.72 + 1.85i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-5.94 - 8.17i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-10.0 + 13.8i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-6.78 + 2.20i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (10.0 + 13.7i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−11.34491670486414012257778798565, −10.66028441294112253165680828376, −9.858269729528746308948225228539, −8.961074196566697608277253748719, −8.321267977351043932653470849178, −6.24599087892954577693192494794, −5.41696083978087891338383709733, −4.13109731364490015306772252839, −3.09175847773372497511225467934, −1.79765721125305987378575355770,
2.12645392870468343522976983454, 3.47999700913433487780067778886, 4.94664592933379636181178739478, 6.23716083656895005783617201183, 6.86436817423233933995073685704, 7.898402988419719477244253624657, 8.896790155017873876531782473828, 9.601971950566241260013289811850, 10.93813115422259123564408426162, 12.47814541437040360569703120935