L(s) = 1 | + (−0.0499 − 0.186i)2-s + (0.806 + 1.53i)3-s + (1.69 − 0.981i)4-s + (0.245 − 0.226i)6-s + (2.35 − 0.632i)7-s + (−0.540 − 0.540i)8-s + (−1.69 + 2.47i)9-s + (−2.14 − 1.23i)11-s + (2.87 + 1.81i)12-s + (−1.57 − 0.422i)13-s + (−0.235 − 0.407i)14-s + (1.88 − 3.27i)16-s + (0.403 − 0.403i)17-s + (0.545 + 0.193i)18-s + 4.28i·19-s + ⋯ |
L(s) = 1 | + (−0.0352 − 0.131i)2-s + (0.465 + 0.885i)3-s + (0.849 − 0.490i)4-s + (0.100 − 0.0925i)6-s + (0.891 − 0.238i)7-s + (−0.191 − 0.191i)8-s + (−0.566 + 0.823i)9-s + (−0.646 − 0.373i)11-s + (0.829 + 0.523i)12-s + (−0.436 − 0.117i)13-s + (−0.0629 − 0.108i)14-s + (0.472 − 0.818i)16-s + (0.0979 − 0.0979i)17-s + (0.128 + 0.0455i)18-s + 0.983i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.62323 + 0.195970i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.62323 + 0.195970i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.806 - 1.53i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (0.0499 + 0.186i)T + (-1.73 + i)T^{2} \) |
| 7 | \( 1 + (-2.35 + 0.632i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (2.14 + 1.23i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.57 + 0.422i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (-0.403 + 0.403i)T - 17iT^{2} \) |
| 19 | \( 1 - 4.28iT - 19T^{2} \) |
| 23 | \( 1 + (1.82 - 6.82i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-3.20 + 5.55i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.97 + 3.41i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.171 - 0.171i)T + 37iT^{2} \) |
| 41 | \( 1 + (6.52 - 3.76i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.32 + 4.95i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (0.780 + 2.91i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (6.12 + 6.12i)T + 53iT^{2} \) |
| 59 | \( 1 + (-2.27 - 3.93i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.235 - 0.408i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.443 + 1.65i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 3.50iT - 71T^{2} \) |
| 73 | \( 1 + (6.88 - 6.88i)T - 73iT^{2} \) |
| 79 | \( 1 + (-6.50 - 3.75i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-10.6 + 2.85i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + 2.90T + 89T^{2} \) |
| 97 | \( 1 + (1.41 - 0.379i)T + (84.0 - 48.5i)T^{2} \) |
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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
−11.87981678781816878604583861588, −11.22606370587233991937008004329, −10.29058268029836775133718977814, −9.707108203039476493620385007661, −8.226100561788149117957428449200, −7.54368837583964353025611205039, −5.87310842173005029915120435566, −4.95145637108212669452364018885, −3.45236381591024704199432257631, −2.02997163679280467484374166194,
1.93655876452131087457040210859, 2.98225350498413520083731061042, 4.90462267882424870767927332485, 6.41193247333960357199991319570, 7.26815327302698981897974590524, 8.098676965365665567036222256731, 8.863474328116765502145413909386, 10.48052597950914625844010519380, 11.44973297632380393903234591079, 12.30587393709643030445870961922