L(s) = 1 | + (−1.5 − 2.59i)3-s + (0.5 − 0.866i)5-s + (−3 + 5.19i)9-s + (2.5 + 4.33i)11-s − 3·13-s − 3·15-s + (0.5 + 0.866i)17-s + (−3 + 5.19i)19-s + (−3 + 5.19i)23-s + (−0.499 − 0.866i)25-s + 9·27-s − 9·29-s + (2 + 3.46i)31-s + (7.50 − 12.9i)33-s + (−1 + 1.73i)37-s + ⋯ |
L(s) = 1 | + (−0.866 − 1.49i)3-s + (0.223 − 0.387i)5-s + (−1 + 1.73i)9-s + (0.753 + 1.30i)11-s − 0.832·13-s − 0.774·15-s + (0.121 + 0.210i)17-s + (−0.688 + 1.19i)19-s + (−0.625 + 1.08i)23-s + (−0.0999 − 0.173i)25-s + 1.73·27-s − 1.67·29-s + (0.359 + 0.622i)31-s + (1.30 − 2.26i)33-s + (−0.164 + 0.284i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.526756 + 0.261127i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.526756 + 0.261127i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (1.5 + 2.59i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.5 - 11.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + (2 - 3.46i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 13T + 97T^{2} \) |
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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
−10.10029021743968538023671421925, −9.374713437772091599844849458417, −8.140181850176752005969610980992, −7.44993758185122295633808126641, −6.79469758201643593411032712868, −5.91614061787608399566901158135, −5.19033671783751511397496297411, −3.99935389507306078874350177370, −2.09494057477336641200053054371, −1.48724060963417655401236893420,
0.30127134837684575171694354925, 2.64709451011125062993507827783, 3.81359052049937885899183199778, 4.52325463981775265974686942944, 5.58222151954653710203048705420, 6.14958816115782488649017942668, 7.13796872600580535725060506007, 8.563787333573754041189024056033, 9.267628747341843134584118262387, 9.934430994249639686033959697071