L(s) = 1 | + (−0.309 − 0.951i)2-s + (−1.69 − 1.23i)3-s + (−0.809 + 0.587i)4-s + (0.760 − 2.33i)5-s + (−0.649 + 1.99i)6-s + (2.50 − 1.82i)7-s + (0.809 + 0.587i)8-s + (0.436 + 1.34i)9-s − 2.46·10-s + (3.02 − 1.36i)11-s + 2.10·12-s + (−1.22 − 3.78i)13-s + (−2.50 − 1.82i)14-s + (−4.18 + 3.03i)15-s + (0.309 − 0.951i)16-s + (−0.663 + 2.04i)17-s + ⋯ |
L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.981 − 0.712i)3-s + (−0.404 + 0.293i)4-s + (0.339 − 1.04i)5-s + (−0.265 + 0.815i)6-s + (0.947 − 0.688i)7-s + (0.286 + 0.207i)8-s + (0.145 + 0.448i)9-s − 0.777·10-s + (0.912 − 0.410i)11-s + 0.606·12-s + (−0.340 − 1.04i)13-s + (−0.670 − 0.486i)14-s + (−1.07 + 0.784i)15-s + (0.0772 − 0.237i)16-s + (−0.160 + 0.494i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0290120 - 0.925185i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0290120 - 0.925185i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 11 | \( 1 + (-3.02 + 1.36i)T \) |
| 19 | \( 1 + (-0.809 - 0.587i)T \) |
good | 3 | \( 1 + (1.69 + 1.23i)T + (0.927 + 2.85i)T^{2} \) |
| 5 | \( 1 + (-0.760 + 2.33i)T + (-4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (-2.50 + 1.82i)T + (2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (1.22 + 3.78i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.663 - 2.04i)T + (-13.7 - 9.99i)T^{2} \) |
| 23 | \( 1 + 2.55T + 23T^{2} \) |
| 29 | \( 1 + (5.26 - 3.82i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.459 - 1.41i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.41 + 1.02i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (4.43 + 3.22i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 4.59T + 43T^{2} \) |
| 47 | \( 1 + (-2.75 - 2.00i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (3.63 + 11.1i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-5.38 + 3.90i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (1.95 - 6.03i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 + (4.93 - 15.1i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-2.56 + 1.86i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.24 - 6.90i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.46 + 10.6i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 4.63T + 89T^{2} \) |
| 97 | \( 1 + (1.69 + 5.20i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.03914431596421133156192954845, −10.04150499957512801260498234381, −8.935701051028451626125231853300, −8.100488341660568863600864233336, −7.07703730334696660518952001147, −5.76954654999295853691811762803, −5.02243355524755163103217775080, −3.78974505599213232350647683528, −1.65184703520360715659472219547, −0.78603850719315197454633668167,
2.13634850328721266566042569101, 4.17032405935729145759211406819, 5.03838433061437983196216226021, 6.03052516242555537325177617903, 6.76497513640786096970439058823, 7.81876393192418127684558789936, 9.153123761281986666837075325081, 9.768989974794140247863834556908, 10.79423725514712879064865090472, 11.50008216755262564953808323530