L(s) = 1 | + (−12.0 + 12.0i)3-s + (−34.4 + 44.0i)5-s + (74.8 + 74.8i)7-s − 49.4i·9-s + 432. i·11-s + (639. + 639. i)13-s + (−116. − 948. i)15-s + (−1.42e3 + 1.42e3i)17-s + 2.37e3·19-s − 1.81e3·21-s + (−170. + 170. i)23-s + (−757. − 3.03e3i)25-s + (−2.34e3 − 2.34e3i)27-s + 5.00e3i·29-s − 7.16e3i·31-s + ⋯ |
L(s) = 1 | + (−0.775 + 0.775i)3-s + (−0.615 + 0.788i)5-s + (0.577 + 0.577i)7-s − 0.203i·9-s + 1.07i·11-s + (1.04 + 1.04i)13-s + (−0.133 − 1.08i)15-s + (−1.19 + 1.19i)17-s + 1.50·19-s − 0.895·21-s + (−0.0671 + 0.0671i)23-s + (−0.242 − 0.970i)25-s + (−0.617 − 0.617i)27-s + 1.10i·29-s − 1.33i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.346i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.937 + 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.093942915\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.093942915\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (34.4 - 44.0i)T \) |
good | 3 | \( 1 + (12.0 - 12.0i)T - 243iT^{2} \) |
| 7 | \( 1 + (-74.8 - 74.8i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 - 432. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-639. - 639. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (1.42e3 - 1.42e3i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 - 2.37e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (170. - 170. i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 - 5.00e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 7.16e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (4.64e3 - 4.64e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 - 1.57e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + (-4.52e3 + 4.52e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (489. + 489. i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (1.53e3 + 1.53e3i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + 2.54e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.14e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-3.79e4 - 3.79e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 + 4.14e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-3.76e4 - 3.76e4i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 + 3.97e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-4.03e4 + 4.03e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 - 4.83e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-6.01e4 + 6.01e4i)T - 8.58e9iT^{2} \) |
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
−12.12292048107782721834515211044, −11.35041618683982331790386925104, −10.81124713140333040460324909380, −9.688431636383245091124321351797, −8.470291145050974862674875323681, −7.19157563397825844201425599263, −6.05273163024138579647024767582, −4.76381788620134135560201382991, −3.83212184033204187763820644588, −1.94622023970703921396831642532,
0.48873541433439014648581226924, 1.09940717546342320374678929161, 3.43904121483544034575186496967, 4.93900456035530662227614043345, 5.94829876931895347398655550441, 7.24758851622928888969667702041, 8.128364340015709120401735464610, 9.181745142144987500476775265530, 10.93075647810485478041462308033, 11.41452904048857113543424362113