L(s) = 1 | + (1.12 − 0.820i)2-s + (−0.511 − 1.57i)3-s + (−0.0158 + 0.0488i)4-s + (0.809 + 0.587i)5-s + (−1.86 − 1.35i)6-s + (−1.45 + 4.47i)7-s + (0.884 + 2.72i)8-s + (0.209 − 0.152i)9-s + 1.39·10-s + 0.0850·12-s + (4.08 − 2.96i)13-s + (2.03 + 6.24i)14-s + (0.511 − 1.57i)15-s + (3.15 + 2.28i)16-s + (4.29 + 3.12i)17-s + (0.111 − 0.344i)18-s + ⋯ |
L(s) = 1 | + (0.798 − 0.580i)2-s + (−0.295 − 0.908i)3-s + (−0.00793 + 0.0244i)4-s + (0.361 + 0.262i)5-s + (−0.763 − 0.554i)6-s + (−0.549 + 1.69i)7-s + (0.312 + 0.962i)8-s + (0.0699 − 0.0508i)9-s + 0.441·10-s + 0.0245·12-s + (1.13 − 0.823i)13-s + (0.542 + 1.67i)14-s + (0.132 − 0.406i)15-s + (0.787 + 0.572i)16-s + (1.04 + 0.757i)17-s + (0.0263 − 0.0812i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.10994 - 0.259367i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.10994 - 0.259367i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.809 - 0.587i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-1.12 + 0.820i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (0.511 + 1.57i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + (1.45 - 4.47i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (-4.08 + 2.96i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-4.29 - 3.12i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (0.698 + 2.14i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 1.05T + 23T^{2} \) |
| 29 | \( 1 + (0.862 - 2.65i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (3.02 - 2.19i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (0.244 - 0.753i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.90 - 5.85i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 2.70T + 43T^{2} \) |
| 47 | \( 1 + (3.47 + 10.6i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.850 - 0.617i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.40 + 4.31i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-7.99 - 5.80i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + (-0.850 - 0.617i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.12 + 9.60i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (7.61 - 5.53i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (2.55 + 1.85i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 + (-14.3 + 10.4i)T + (29.9 - 92.2i)T^{2} \) |
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
−10.98543075420509290542912937311, −9.869764371650152555456870656749, −8.734293982829253841396357218497, −8.071880169536255585523893102719, −6.76753021510974804345116641747, −5.84170374653189809588069888664, −5.36983840480648481240141551148, −3.64726541503676924649568978200, −2.80263923412488091099660853651, −1.66370848278791963564489097589,
1.11801394251362155253791475136, 3.72308077156115481422480164462, 4.08636241020196632795386622565, 5.08276633159633306699542071028, 5.99665775941910197880380641203, 6.88992479576008680279882689320, 7.71195196584220775578770087875, 9.337822006228208084112251835741, 9.870579728128341942187186470829, 10.55199551970029241116031049165