L(s) = 1 | − 0.129i·2-s − 2.08i·3-s + 1.98·4-s + 1.78i·5-s − 0.270·6-s + (1.65 − 2.06i)7-s − 0.517i·8-s − 1.33·9-s + 0.231·10-s − 4.12i·12-s + 3.79·13-s + (−0.267 − 0.215i)14-s + 3.71·15-s + 3.89·16-s − 5.68·17-s + 0.173i·18-s + ⋯ |
L(s) = 1 | − 0.0918i·2-s − 1.20i·3-s + 0.991·4-s + 0.797i·5-s − 0.110·6-s + (0.626 − 0.779i)7-s − 0.183i·8-s − 0.445·9-s + 0.0733·10-s − 1.19i·12-s + 1.05·13-s + (−0.0716 − 0.0575i)14-s + 0.959·15-s + 0.974·16-s − 1.37·17-s + 0.0408i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.77895 - 1.25227i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.77895 - 1.25227i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-1.65 + 2.06i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 0.129iT - 2T^{2} \) |
| 3 | \( 1 + 2.08iT - 3T^{2} \) |
| 5 | \( 1 - 1.78iT - 5T^{2} \) |
| 13 | \( 1 - 3.79T + 13T^{2} \) |
| 17 | \( 1 + 5.68T + 17T^{2} \) |
| 19 | \( 1 + 4.66T + 19T^{2} \) |
| 23 | \( 1 - 6.35T + 23T^{2} \) |
| 29 | \( 1 - 3.12iT - 29T^{2} \) |
| 31 | \( 1 - 1.29iT - 31T^{2} \) |
| 37 | \( 1 - 0.949T + 37T^{2} \) |
| 41 | \( 1 - 2.70T + 41T^{2} \) |
| 43 | \( 1 + 7.27iT - 43T^{2} \) |
| 47 | \( 1 - 12.8iT - 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 + 8.25iT - 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 - 3.79T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 - 4.61T + 73T^{2} \) |
| 79 | \( 1 + 0.183iT - 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 3.59iT - 89T^{2} \) |
| 97 | \( 1 + 3.79iT - 97T^{2} \) |
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
−10.62354566883052411303606985155, −8.994473496366930551152095507181, −8.014330793973957953159597685831, −7.27915403290513565201769919764, −6.63883593283588666577267041287, −6.23824374747713157992336799951, −4.57935464737669795876080918712, −3.25572393652681184768348006397, −2.15632281122203954434506842836, −1.22027206765137898687660599182,
1.63096542867733290365858962124, 2.89401502479672258867562969429, 4.23321265571675849899879498195, 4.91115162599605449009961552177, 5.89032832349640638139235403827, 6.75121384251067057383718304083, 8.083798058888162510331203384840, 8.808666099602776941281236885396, 9.321693095102349453027353025304, 10.66936441524888948185173549439