L(s) = 1 | + (2.59 + 1.5i)3-s + (1.73 + 2i)7-s + (3 + 5.19i)9-s + (0.5 − 0.866i)11-s − 2i·13-s + (−2.59 − 1.5i)17-s + (2.5 + 4.33i)19-s + (1.5 + 7.79i)21-s + (2.59 − 1.5i)23-s + 9i·27-s + 6·29-s + (0.5 − 0.866i)31-s + (2.59 − 1.5i)33-s + (−4.33 + 2.5i)37-s + (3 − 5.19i)39-s + ⋯ |
L(s) = 1 | + (1.49 + 0.866i)3-s + (0.654 + 0.755i)7-s + (1 + 1.73i)9-s + (0.150 − 0.261i)11-s − 0.554i·13-s + (−0.630 − 0.363i)17-s + (0.573 + 0.993i)19-s + (0.327 + 1.70i)21-s + (0.541 − 0.312i)23-s + 1.73i·27-s + 1.11·29-s + (0.0898 − 0.155i)31-s + (0.452 − 0.261i)33-s + (−0.711 + 0.410i)37-s + (0.480 − 0.832i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.077758044\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.077758044\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.73 - 2i)T \) |
good | 3 | \( 1 + (-2.59 - 1.5i)T + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + (2.59 + 1.5i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.59 + 1.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.33 - 2.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.79 + 4.5i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.5 + 2.59i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.52 + 5.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + (-6.06 - 3.5i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−9.621338096074645903857884652626, −8.794584773090210978916035046766, −8.331350464505880152302142327411, −7.72095486495764131996988297336, −6.49682826297776569943999222570, −5.22996115172301160884515969663, −4.62189654409106141935307866709, −3.47061251594437946512355006275, −2.81698045225542291486480440355, −1.73749201477878856878589822295,
1.17823540563910402632304150023, 2.09541406879392420299476405459, 3.13553626190755665717283993062, 4.09117886390434396199635671697, 5.01143669925527556259083708861, 6.66382406787934641398845607328, 7.03810457015652880763639097212, 7.81207659145143750155855698150, 8.600194310859169897894001267187, 9.094044549603208368503540009519