L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.566 − 0.981i)5-s + (−0.203 − 2.63i)7-s + 0.999i·8-s + (0.981 + 0.566i)10-s + (2.97 + 1.71i)11-s − 1.17i·13-s + (1.49 + 2.18i)14-s + (−0.5 − 0.866i)16-s + (−0.471 + 0.817i)17-s + (0.866 − 0.5i)19-s − 1.13·20-s − 3.43·22-s + (3.02 − 1.74i)23-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.253 − 0.438i)5-s + (−0.0769 − 0.997i)7-s + 0.353i·8-s + (0.310 + 0.179i)10-s + (0.896 + 0.517i)11-s − 0.326i·13-s + (0.399 + 0.583i)14-s + (−0.125 − 0.216i)16-s + (−0.114 + 0.198i)17-s + (0.198 − 0.114i)19-s − 0.253·20-s − 0.731·22-s + (0.631 − 0.364i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.291 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.291 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.160789046\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.160789046\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.203 + 2.63i)T \) |
| 19 | \( 1 + (-0.866 + 0.5i)T \) |
good | 5 | \( 1 + (0.566 + 0.981i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.97 - 1.71i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.17iT - 13T^{2} \) |
| 17 | \( 1 + (0.471 - 0.817i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-3.02 + 1.74i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 0.191iT - 29T^{2} \) |
| 31 | \( 1 + (3.45 + 1.99i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.23 - 2.14i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 1.61T + 43T^{2} \) |
| 47 | \( 1 + (-5.74 - 9.95i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.09 + 3.52i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.890 - 1.54i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7.52 - 4.34i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.96 + 12.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6.68iT - 71T^{2} \) |
| 73 | \( 1 + (4.41 + 2.54i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.39 + 7.60i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 + (-1.75 - 3.04i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 6.99iT - 97T^{2} \) |
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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
−8.924241450965766973700241924496, −7.898720581156773593061863422491, −7.43435313139285391437571300891, −6.62170196365520978780492446113, −5.91414672794206858120470223315, −4.66575915236916553423100389246, −4.19689321240767180409817818367, −2.96245394492760372657509309509, −1.53276764147172218257641283573, −0.56598732576400602542016263534,
1.17354494934459513218392727106, 2.36910882583362723580115437932, 3.22192584858395843698759228404, 4.04999584417952785264564964265, 5.30235660966571432069626802932, 6.10866140214169668968785403896, 6.96038801560436404827007006088, 7.59125834940267504578565461369, 8.664926086342799173490029869612, 9.060933557598902089811618886406