L(s) = 1 | + (0.613 − 1.27i)2-s + (2.05 − 0.551i)3-s + (−1.24 − 1.56i)4-s + (0.929 − 3.47i)5-s + (0.560 − 2.95i)6-s + (−2.75 + 0.627i)8-s + (1.33 − 0.768i)9-s + (−3.85 − 3.31i)10-s + (2.73 − 0.732i)11-s + (−3.42 − 2.53i)12-s + (−1.17 + 1.17i)13-s − 7.65i·15-s + (−0.893 + 3.89i)16-s + (5.31 + 3.06i)17-s + (−0.162 − 2.16i)18-s + (−0.358 + 1.33i)19-s + ⋯ |
L(s) = 1 | + (0.434 − 0.900i)2-s + (1.18 − 0.318i)3-s + (−0.623 − 0.782i)4-s + (0.415 − 1.55i)5-s + (0.228 − 1.20i)6-s + (−0.975 + 0.221i)8-s + (0.443 − 0.256i)9-s + (−1.21 − 1.04i)10-s + (0.823 − 0.220i)11-s + (−0.989 − 0.730i)12-s + (−0.326 + 0.326i)13-s − 1.97i·15-s + (−0.223 + 0.974i)16-s + (1.28 + 0.743i)17-s + (−0.0381 − 0.510i)18-s + (−0.0822 + 0.307i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.861136 - 2.65283i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.861136 - 2.65283i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.613 + 1.27i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-2.05 + 0.551i)T + (2.59 - 1.5i)T^{2} \) |
| 5 | \( 1 + (-0.929 + 3.47i)T + (-4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-2.73 + 0.732i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (1.17 - 1.17i)T - 13iT^{2} \) |
| 17 | \( 1 + (-5.31 - 3.06i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.358 - 1.33i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (0.103 + 0.179i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.46 + 3.46i)T + 29iT^{2} \) |
| 31 | \( 1 + (2.87 - 4.97i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.349 - 0.0935i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 6.75T + 41T^{2} \) |
| 43 | \( 1 + (-0.207 - 0.207i)T + 43iT^{2} \) |
| 47 | \( 1 + (-5.46 - 9.46i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.63 + 9.82i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (0.743 + 2.77i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (10.2 + 2.74i)T + (52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (-0.358 - 1.33i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 2.09T + 71T^{2} \) |
| 73 | \( 1 + (-1.36 + 2.36i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.55 + 5.51i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.27 + 2.27i)T + 83iT^{2} \) |
| 89 | \( 1 + (-7.04 - 12.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 2.83iT - 97T^{2} \) |
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
−9.546196188005505252382962804865, −9.322909040297649870553863369835, −8.499135109549702879834760299610, −7.77847021209698545054703300680, −6.13405147665690494951476368533, −5.28916400617039398307696066554, −4.22172387403086823931972601454, −3.37156383094700349477370319466, −2.00815206254421546517499111374, −1.21498286425763133306986875992,
2.50337330974950583291327535637, 3.25523702810917088820187335416, 4.02342899509103671280120798851, 5.47866294820440933569889933058, 6.35786777597787247452775323553, 7.37634193205384881980902793280, 7.68823113323335645122619846109, 9.034991066587288305658313022139, 9.485794796839805669125985237441, 10.35590715640588091958510501340