L(s) = 1 | + (−1 − i)2-s + (−1.22 + 1.22i)3-s + 2i·4-s + (4.77 − 1.46i)5-s + 2.44·6-s + (8.57 + 8.57i)7-s + (2 − 2i)8-s − 2.99i·9-s + (−6.24 − 3.31i)10-s + 5.33·11-s + (−2.44 − 2.44i)12-s + (9.44 − 9.44i)13-s − 17.1i·14-s + (−4.05 + 7.65i)15-s − 4·16-s + (15.8 + 15.8i)17-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s + (0.955 − 0.293i)5-s + 0.408·6-s + (1.22 + 1.22i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (−0.624 − 0.331i)10-s + 0.485·11-s + (−0.204 − 0.204i)12-s + (0.726 − 0.726i)13-s − 1.22i·14-s + (−0.270 + 0.510i)15-s − 0.250·16-s + (0.932 + 0.932i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0663i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.859881992\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.859881992\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 3 | \( 1 + (1.22 - 1.22i)T \) |
| 5 | \( 1 + (-4.77 + 1.46i)T \) |
| 23 | \( 1 + (-3.39 + 3.39i)T \) |
good | 7 | \( 1 + (-8.57 - 8.57i)T + 49iT^{2} \) |
| 11 | \( 1 - 5.33T + 121T^{2} \) |
| 13 | \( 1 + (-9.44 + 9.44i)T - 169iT^{2} \) |
| 17 | \( 1 + (-15.8 - 15.8i)T + 289iT^{2} \) |
| 19 | \( 1 + 27.9iT - 361T^{2} \) |
| 29 | \( 1 - 43.2iT - 841T^{2} \) |
| 31 | \( 1 + 45.6T + 961T^{2} \) |
| 37 | \( 1 + (22.6 + 22.6i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 27.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + (2.61 - 2.61i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (60.8 + 60.8i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-9.28 + 9.28i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 62.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 100.T + 3.72e3T^{2} \) |
| 67 | \( 1 + (23.2 + 23.2i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 73.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-74.1 + 74.1i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 42.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-30.3 + 30.3i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 115. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-131. - 131. i)T + 9.40e3iT^{2} \) |
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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
−10.47689940444351585928314793866, −9.191146696671191552828476474308, −8.914203615350314075938287947004, −8.031449099429020454227414446249, −6.62712512250163175555815407827, −5.47072962425881436124034488010, −5.09904915575077389872962288526, −3.55236239278684456512016198726, −2.18423133442150373829198538378, −1.18282651698702998834083362361,
1.09904163002360249190822266780, 1.82084327291995461890107083457, 3.84326762917728451924496609643, 5.06028134545651523025438106183, 5.93390792136911950853580026171, 6.80797317574864978613424934614, 7.57606180859814162934894536380, 8.309983718153122528731061106333, 9.548786367879561751008427819559, 10.13268139619141479953177140129