L(s) = 1 | + (1.29 − 2.24i)3-s + (−0.833 − 1.44i)5-s + (2.29 + 1.31i)7-s + (−1.86 − 3.23i)9-s + (−0.0726 + 0.125i)11-s − 1.47·13-s − 4.32·15-s + (2.95 − 5.11i)17-s + (−0.5 − 0.866i)19-s + (5.93 − 3.45i)21-s + (−3.11 − 5.38i)23-s + (1.11 − 1.92i)25-s − 1.90·27-s − 3.13·29-s + (0.441 − 0.764i)31-s + ⋯ |
L(s) = 1 | + (0.749 − 1.29i)3-s + (−0.372 − 0.645i)5-s + (0.867 + 0.497i)7-s + (−0.622 − 1.07i)9-s + (−0.0219 + 0.0379i)11-s − 0.410·13-s − 1.11·15-s + (0.716 − 1.24i)17-s + (−0.114 − 0.198i)19-s + (1.29 − 0.752i)21-s + (−0.648 − 1.12i)23-s + (0.222 − 0.385i)25-s − 0.367·27-s − 0.582·29-s + (0.0792 − 0.137i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.551 + 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.551 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.971638637\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.971638637\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.29 - 1.31i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
good | 3 | \( 1 + (-1.29 + 2.24i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.833 + 1.44i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.0726 - 0.125i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 1.47T + 13T^{2} \) |
| 17 | \( 1 + (-2.95 + 5.11i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (3.11 + 5.38i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 3.13T + 29T^{2} \) |
| 31 | \( 1 + (-0.441 + 0.764i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.981 - 1.69i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 2.05T + 43T^{2} \) |
| 47 | \( 1 + (-0.0872 - 0.151i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.21 - 7.30i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.790 + 1.36i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.27 - 7.40i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.669 - 1.15i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 7.37T + 71T^{2} \) |
| 73 | \( 1 + (2.36 - 4.09i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.74 + 9.95i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4.37T + 83T^{2} \) |
| 89 | \( 1 + (-0.696 - 1.20i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−9.256729572846891732442677982589, −8.586410667918487532296481502257, −7.932821088903223865027877361531, −7.40196145400240805681219639472, −6.41564594242455745652093865977, −5.25448386793075959978218081073, −4.41944339590512299489226775913, −2.89315005764459920489384615764, −2.05295634683614064456578446369, −0.831861757288346735404092970754,
1.86228881271476180327410535785, 3.34734354319176991070742118621, 3.80423528141817234496538340045, 4.76363162008116793505730890769, 5.70003302483540545353641465325, 7.06337620969090089436591834658, 7.924028291109216925857862022064, 8.442758979298090508112932868459, 9.562141121457818284766343166384, 10.10551189720161222930686310912