| L(s) = 1 | + (−1.29 − 0.558i)2-s + (1.06 − 1.36i)3-s + (1.37 + 1.45i)4-s + (1.51 − 1.64i)5-s + (−2.14 + 1.18i)6-s − 1.88i·7-s + (−0.975 − 2.65i)8-s + (−0.737 − 2.90i)9-s + (−2.88 + 1.28i)10-s + (−3.00 + 2.18i)11-s + (3.44 − 0.336i)12-s + (1.87 + 1.36i)13-s + (−1.05 + 2.45i)14-s + (−0.633 − 3.82i)15-s + (−0.215 + 3.99i)16-s + (5.42 − 1.76i)17-s + ⋯ |
| L(s) = 1 | + (−0.918 − 0.395i)2-s + (0.614 − 0.789i)3-s + (0.687 + 0.725i)4-s + (0.678 − 0.734i)5-s + (−0.875 + 0.482i)6-s − 0.714i·7-s + (−0.344 − 0.938i)8-s + (−0.245 − 0.969i)9-s + (−0.913 + 0.406i)10-s + (−0.906 + 0.658i)11-s + (0.995 − 0.0971i)12-s + (0.520 + 0.378i)13-s + (−0.282 + 0.656i)14-s + (−0.163 − 0.986i)15-s + (−0.0539 + 0.998i)16-s + (1.31 − 0.427i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.319 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.653270 - 0.909929i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.653270 - 0.909929i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.29 + 0.558i)T \) |
| 3 | \( 1 + (-1.06 + 1.36i)T \) |
| 5 | \( 1 + (-1.51 + 1.64i)T \) |
| good | 7 | \( 1 + 1.88iT - 7T^{2} \) |
| 11 | \( 1 + (3.00 - 2.18i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-1.87 - 1.36i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-5.42 + 1.76i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.894 + 0.290i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (4.58 - 3.32i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (4.43 + 1.43i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.681 + 0.221i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-3.22 - 2.34i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (4.79 - 6.60i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 1.32iT - 43T^{2} \) |
| 47 | \( 1 + (-2.90 + 8.95i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-5.10 - 1.65i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-9.81 - 7.12i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-4.71 + 3.42i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-7.58 + 2.46i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.63 - 5.04i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (11.4 - 8.34i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (5.35 + 1.74i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.67 - 11.3i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-9.99 - 13.7i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (3.00 - 9.25i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.56966112919115319677734793497, −10.07787771550454849156781324189, −9.714346234677587852405158712252, −8.552757975743338508974397720463, −7.78707724302249315210226614551, −6.99591358695851766843012327434, −5.65900190739378599075307218435, −3.80573571734135925994003218390, −2.31843786264058128790415521542, −1.11539188360011740277301889048,
2.22392135602443521811201672454, 3.29597833415161355565311429176, 5.47006476524894226697280733525, 5.93837176337736361261537224548, 7.53882329273480350411372793405, 8.338526880427209380596365066223, 9.181684518501445652973369433719, 10.23094795986781275517241818676, 10.51352496580311711283130502256, 11.60632597573046699130540409679
