L(s) = 1 | + 2.85i·3-s + (−1.43 + 1.71i)5-s + (0.458 − 0.458i)7-s − 5.15·9-s + (−0.492 + 0.492i)11-s − 4.52·13-s + (−4.89 − 4.09i)15-s + (−3.12 + 3.12i)17-s + (4.04 − 4.04i)19-s + (1.31 + 1.31i)21-s + (1.80 + 1.80i)23-s + (−0.881 − 4.92i)25-s − 6.15i·27-s + (3.83 + 3.83i)29-s + 0.139i·31-s + ⋯ |
L(s) = 1 | + 1.64i·3-s + (−0.641 + 0.766i)5-s + (0.173 − 0.173i)7-s − 1.71·9-s + (−0.148 + 0.148i)11-s − 1.25·13-s + (−1.26 − 1.05i)15-s + (−0.758 + 0.758i)17-s + (0.928 − 0.928i)19-s + (0.285 + 0.285i)21-s + (0.376 + 0.376i)23-s + (−0.176 − 0.984i)25-s − 1.18i·27-s + (0.712 + 0.712i)29-s + 0.0251i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.179930 - 0.713979i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.179930 - 0.713979i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.43 - 1.71i)T \) |
good | 3 | \( 1 - 2.85iT - 3T^{2} \) |
| 7 | \( 1 + (-0.458 + 0.458i)T - 7iT^{2} \) |
| 11 | \( 1 + (0.492 - 0.492i)T - 11iT^{2} \) |
| 13 | \( 1 + 4.52T + 13T^{2} \) |
| 17 | \( 1 + (3.12 - 3.12i)T - 17iT^{2} \) |
| 19 | \( 1 + (-4.04 + 4.04i)T - 19iT^{2} \) |
| 23 | \( 1 + (-1.80 - 1.80i)T + 23iT^{2} \) |
| 29 | \( 1 + (-3.83 - 3.83i)T + 29iT^{2} \) |
| 31 | \( 1 - 0.139iT - 31T^{2} \) |
| 37 | \( 1 + 5.84T + 37T^{2} \) |
| 41 | \( 1 + 4.55iT - 41T^{2} \) |
| 43 | \( 1 + 7.49T + 43T^{2} \) |
| 47 | \( 1 + (-4.14 - 4.14i)T + 47iT^{2} \) |
| 53 | \( 1 + 2.75iT - 53T^{2} \) |
| 59 | \( 1 + (3.62 + 3.62i)T + 59iT^{2} \) |
| 61 | \( 1 + (3.72 - 3.72i)T - 61iT^{2} \) |
| 67 | \( 1 - 3.32T + 67T^{2} \) |
| 71 | \( 1 + 1.37T + 71T^{2} \) |
| 73 | \( 1 + (2.55 - 2.55i)T - 73iT^{2} \) |
| 79 | \( 1 - 3.86T + 79T^{2} \) |
| 83 | \( 1 - 14.4iT - 83T^{2} \) |
| 89 | \( 1 + 3.35T + 89T^{2} \) |
| 97 | \( 1 + (4.95 - 4.95i)T - 97iT^{2} \) |
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
−10.85116032131889130506448586914, −10.32262840674642007638825502664, −9.503914574161298581057408511615, −8.688194752601963526616780755626, −7.56892008967107727553239274287, −6.70550517265486631400709705869, −5.23192172535350987735148907002, −4.59298487646303949386196022217, −3.60839464740730455726339199995, −2.68943770901227998069785373143,
0.38784597263030990389281731674, 1.77206318595329899296613565945, 2.99002118778259211697633113358, 4.65624353036751989305736424652, 5.54759303630207946035600778549, 6.76372046078297927705197408709, 7.45641946237273144692937792276, 8.125174654082322187398124303294, 8.891240663426026533120321046847, 9.993904452351200224556033284900