L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.793 + 0.608i)3-s + (0.499 + 0.866i)4-s + (−0.991 + 0.130i)6-s + 0.999i·8-s + (0.258 − 0.965i)9-s + (−0.448 − 0.258i)11-s + (−0.923 − 0.382i)12-s + (−0.5 + 0.866i)16-s − 1.98·17-s + (0.707 − 0.707i)18-s + 1.58i·19-s + (−0.258 − 0.448i)22-s + (−0.608 − 0.793i)24-s + (−0.5 + 0.866i)25-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.793 + 0.608i)3-s + (0.499 + 0.866i)4-s + (−0.991 + 0.130i)6-s + 0.999i·8-s + (0.258 − 0.965i)9-s + (−0.448 − 0.258i)11-s + (−0.923 − 0.382i)12-s + (−0.5 + 0.866i)16-s − 1.98·17-s + (0.707 − 0.707i)18-s + 1.58i·19-s + (−0.258 − 0.448i)22-s + (−0.608 − 0.793i)24-s + (−0.5 + 0.866i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.077029922\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.077029922\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.793 - 0.608i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + 1.98T + T^{2} \) |
| 19 | \( 1 - 1.58iT - T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.130 + 0.226i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (-0.608 - 1.05i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + 1.21iT - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - 1.84T + T^{2} \) |
| 97 | \( 1 + (0.226 + 0.130i)T + (0.5 + 0.866i)T^{2} \) |
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
−8.998533124361548697676237513612, −8.292881696418312391619742353075, −7.42679083627519369276291094154, −6.60826962574115090898007349615, −6.02950680851332723397946669553, −5.36958661709534005077741266073, −4.58268017700945694367887500753, −3.95425702767837524542230511472, −3.11346595661692576914119106311, −1.86501440951890565720959198219,
0.47599565512967195702798966557, 2.05049025907797447877295683635, 2.46165484513732746269842100542, 3.84273887698934728590329264334, 4.83276508774292458981608391775, 5.06095411439631144196945091071, 6.22425227832006943817623980599, 6.69041649964851577070800647531, 7.26254755566190090438517479398, 8.339040596568676480541981882405