L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.965 − 0.258i)3-s + (−0.866 + 0.499i)4-s − i·6-s + (1.52 + 2.15i)7-s + (−0.707 − 0.707i)8-s + (0.866 + 0.499i)9-s + (0.883 + 1.52i)11-s + (0.965 − 0.258i)12-s + (−2.71 + 2.71i)13-s + (−1.69 + 2.03i)14-s + (0.500 − 0.866i)16-s + (0.574 − 2.14i)17-s + (−0.258 + 0.965i)18-s + (0.886 − 1.53i)19-s + ⋯ |
L(s) = 1 | + (0.183 + 0.683i)2-s + (−0.557 − 0.149i)3-s + (−0.433 + 0.249i)4-s − 0.408i·6-s + (0.577 + 0.816i)7-s + (−0.249 − 0.249i)8-s + (0.288 + 0.166i)9-s + (0.266 + 0.461i)11-s + (0.278 − 0.0747i)12-s + (−0.752 + 0.752i)13-s + (−0.451 + 0.543i)14-s + (0.125 − 0.216i)16-s + (0.139 − 0.520i)17-s + (−0.0610 + 0.227i)18-s + (0.203 − 0.352i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.863 - 0.503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.863 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.103273610\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.103273610\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 3 | \( 1 + (0.965 + 0.258i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.52 - 2.15i)T \) |
good | 11 | \( 1 + (-0.883 - 1.52i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.71 - 2.71i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.574 + 2.14i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.886 + 1.53i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.90 + 1.04i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 3.84iT - 29T^{2} \) |
| 31 | \( 1 + (8.94 - 5.16i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.861 - 3.21i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 11.8iT - 41T^{2} \) |
| 43 | \( 1 + (3.46 + 3.46i)T + 43iT^{2} \) |
| 47 | \( 1 + (5.93 - 1.59i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (0.106 - 0.396i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-5.18 - 8.97i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.87 + 3.39i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.37 - 1.97i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + (10.2 + 2.75i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (10.9 + 6.34i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.94 + 1.94i)T - 83iT^{2} \) |
| 89 | \( 1 + (0.558 - 0.966i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.26 - 7.26i)T + 97iT^{2} \) |
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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
−10.13859380590569996733979601180, −9.228773565103525094482303907866, −8.658351001308217235868220402834, −7.45351937692670388672609445666, −6.97638087818268436654669876658, −6.00878988674747290815000300112, −4.99128330849254717365739842729, −4.66371556492397404160358067015, −3.05347235481771540499041450922, −1.63533537418625158125020099659,
0.51443863736795240421692658149, 1.85085545169642869027110107295, 3.36595741452718448294634323111, 4.17767958014350161657877029347, 5.18042082517450261803387907399, 5.86094398487392349287508437238, 7.13235756631481505762844848458, 7.86242609243097203358770952975, 8.911311722286820824186183721848, 9.892746649135736190629811553883