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 | \( 1 + (-6.01e4 - 6.01e4i)T + 8.58e9iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.41452904048857113543424362113, −10.93075647810485478041462308033, −9.181745142144987500476775265530, −8.128364340015709120401735464610, −7.24758851622928888969667702041, −5.94829876931895347398655550441, −4.93900456035530662227614043345, −3.43904121483544034575186496967, −1.09940717546342320374678929161, −0.48873541433439014648581226924,
1.94622023970703921396831642532, 3.83212184033204187763820644588, 4.76381788620134135560201382991, 6.05273163024138579647024767582, 7.19157563397825844201425599263, 8.470291145050974862674875323681, 9.688431636383245091124321351797, 10.81124713140333040460324909380, 11.35041618683982331790386925104, 12.12292048107782721834515211044