L(s) = 1 | + (1.03 − 0.960i)2-s + (−0.219 − 0.819i)3-s + (0.155 − 1.99i)4-s + (0.356 − 1.33i)5-s + (−1.01 − 0.640i)6-s + (−1.75 − 2.21i)8-s + (1.97 − 1.14i)9-s + (−0.908 − 1.72i)10-s + (2.35 − 0.631i)11-s + (−1.66 + 0.310i)12-s + (1.90 − 1.90i)13-s − 1.16·15-s + (−3.95 − 0.621i)16-s + (−3.35 + 5.81i)17-s + (0.955 − 3.07i)18-s + (−4.02 − 1.07i)19-s + ⋯ |
L(s) = 1 | + (0.734 − 0.679i)2-s + (−0.126 − 0.473i)3-s + (0.0778 − 0.996i)4-s + (0.159 − 0.595i)5-s + (−0.414 − 0.261i)6-s + (−0.619 − 0.784i)8-s + (0.658 − 0.380i)9-s + (−0.287 − 0.545i)10-s + (0.710 − 0.190i)11-s + (−0.481 + 0.0895i)12-s + (0.528 − 0.528i)13-s − 0.302·15-s + (−0.987 − 0.155i)16-s + (−0.814 + 1.41i)17-s + (0.225 − 0.725i)18-s + (−0.924 − 0.247i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.703221 - 2.17544i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.703221 - 2.17544i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.03 + 0.960i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.219 + 0.819i)T + (-2.59 + 1.5i)T^{2} \) |
| 5 | \( 1 + (-0.356 + 1.33i)T + (-4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-2.35 + 0.631i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-1.90 + 1.90i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.35 - 5.81i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.02 + 1.07i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-4.58 + 2.64i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.03 - 3.03i)T - 29iT^{2} \) |
| 31 | \( 1 + (0.599 - 1.03i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.823 - 3.07i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 3.94iT - 41T^{2} \) |
| 43 | \( 1 + (7.02 + 7.02i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.53 - 2.65i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.12 + 1.10i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (6.78 - 1.81i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-13.2 - 3.54i)T + (52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (-1.30 - 4.85i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 11.5iT - 71T^{2} \) |
| 73 | \( 1 + (-8.94 - 5.16i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.03 + 3.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.17 + 9.17i)T - 83iT^{2} \) |
| 89 | \( 1 + (-14.6 + 8.46i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 2.51T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.26166569155563443165773667997, −9.028755181427795861849990575709, −8.587844335057293640044720809770, −6.94020560158070270679503851172, −6.40538854858994215076984775042, −5.40687128708165953801757219222, −4.33743270991948230717807285810, −3.55128258311342954952733369463, −1.97570230516280259492108325187, −0.990655590714848387666652333781,
2.18176192036031720139862944210, 3.50692868987081293655360726143, 4.41486565664821288695749355847, 5.14459705331276084163081550923, 6.47582794687789103173190430475, 6.84516966999658116353747142751, 7.82709451301070520200479244660, 8.995565042237760312415432001175, 9.618412233416978804401510515490, 10.91344640268534170344499076135