L(s) = 1 | + (1.03 + 2.63i)2-s + (5.55 − 5.55i)3-s + (−5.87 + 5.43i)4-s + (−10.4 + 3.84i)5-s + (20.3 + 8.91i)6-s + (−1.14 − 1.14i)7-s + (−20.3 − 9.86i)8-s − 34.8i·9-s + (−20.9 − 23.6i)10-s − 27.0i·11-s + (−2.45 + 62.8i)12-s + (40.4 + 40.4i)13-s + (1.83 − 4.18i)14-s + (−37.0 + 79.7i)15-s + (5.00 − 63.8i)16-s + (−36.2 + 36.2i)17-s + ⋯ |
L(s) = 1 | + (0.364 + 0.931i)2-s + (1.06 − 1.06i)3-s + (−0.734 + 0.678i)4-s + (−0.939 + 0.343i)5-s + (1.38 + 0.606i)6-s + (−0.0616 − 0.0616i)7-s + (−0.899 − 0.436i)8-s − 1.28i·9-s + (−0.662 − 0.749i)10-s − 0.741i·11-s + (−0.0591 + 1.51i)12-s + (0.863 + 0.863i)13-s + (0.0349 − 0.0798i)14-s + (−0.637 + 1.37i)15-s + (0.0781 − 0.996i)16-s + (−0.517 + 0.517i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.33190 + 0.371138i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.33190 + 0.371138i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.03 - 2.63i)T \) |
| 5 | \( 1 + (10.4 - 3.84i)T \) |
good | 3 | \( 1 + (-5.55 + 5.55i)T - 27iT^{2} \) |
| 7 | \( 1 + (1.14 + 1.14i)T + 343iT^{2} \) |
| 11 | \( 1 + 27.0iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (-40.4 - 40.4i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 + (36.2 - 36.2i)T - 4.91e3iT^{2} \) |
| 19 | \( 1 - 56.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + (54.9 - 54.9i)T - 1.21e4iT^{2} \) |
| 29 | \( 1 - 57.1iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 190. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (50.4 - 50.4i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + 71.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + (66.9 - 66.9i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + (343. + 343. i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 + (-240. - 240. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + 738.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 187.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (-576. - 576. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 + 157. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-180. - 180. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + 55.6T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-858. + 858. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + 158. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (1.11e3 - 1.11e3i)T - 9.12e5iT^{2} \) |
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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
−18.24748149426735866095189137084, −16.42025702197973340121498561343, −15.21943901640500128088577515106, −14.04381626368414018450425463278, −13.23476583981247802535681824897, −11.71286980731045095899664950840, −8.790069629864098078359768572483, −7.82474583585897848069012564126, −6.57999626026445557564024249492, −3.59623023422900153389344886929,
3.30343824573709458557378305947, 4.66690016842414864037924154838, 8.323334158758097517523656759505, 9.532623484645942744216758494868, 10.86728320443560279539907719938, 12.42131574827743498282685984776, 13.88775020313447513222167681162, 15.16151145561317545770365985661, 15.89444940685217628237015663753, 18.12948571811663704418260965544