L(s) = 1 | − 1.90i·3-s + (−4.37 − 2.41i)5-s − 3.95·7-s + 5.35·9-s − 6.18·11-s − 4.94·13-s + (−4.60 + 8.35i)15-s + 22.4i·17-s − 10.3·19-s + 7.54i·21-s − 39.6·23-s + (13.3 + 21.1i)25-s − 27.4i·27-s + 30.4i·29-s + 24.7i·31-s + ⋯ |
L(s) = 1 | − 0.636i·3-s + (−0.875 − 0.482i)5-s − 0.565·7-s + 0.595·9-s − 0.562·11-s − 0.380·13-s + (−0.306 + 0.557i)15-s + 1.31i·17-s − 0.546·19-s + 0.359i·21-s − 1.72·23-s + (0.534 + 0.845i)25-s − 1.01i·27-s + 1.04i·29-s + 0.797i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.721 - 0.692i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0149546 + 0.0371528i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0149546 + 0.0371528i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (4.37 + 2.41i)T \) |
good | 3 | \( 1 + 1.90iT - 9T^{2} \) |
| 7 | \( 1 + 3.95T + 49T^{2} \) |
| 11 | \( 1 + 6.18T + 121T^{2} \) |
| 13 | \( 1 + 4.94T + 169T^{2} \) |
| 17 | \( 1 - 22.4iT - 289T^{2} \) |
| 19 | \( 1 + 10.3T + 361T^{2} \) |
| 23 | \( 1 + 39.6T + 529T^{2} \) |
| 29 | \( 1 - 30.4iT - 841T^{2} \) |
| 31 | \( 1 - 24.7iT - 961T^{2} \) |
| 37 | \( 1 + 24.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 31.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 52.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 13.1T + 2.20e3T^{2} \) |
| 53 | \( 1 - 17.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 104.T + 3.48e3T^{2} \) |
| 61 | \( 1 + 57.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 99.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 16.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 96.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 139. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 91.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 94.7T + 7.92e3T^{2} \) |
| 97 | \( 1 + 143. iT - 9.40e3T^{2} \) |
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
−12.20731733434307089888929748089, −10.79006586497478318560855163801, −10.02313652089219074028009479859, −8.718396210646385904574471998915, −7.919863571511493803347209195199, −7.08063589885985584485114858300, −6.02072911036254329483971057952, −4.60652418935913305015261308651, −3.53888745767984038227408704895, −1.77404933418191759674797433764,
0.01829118253767435364588827431, 2.62833269702359924076322423650, 3.87848038995359616507473042669, 4.72693484541851779709617991042, 6.19915084861525754263900868824, 7.31752540918733266562298553883, 8.058232773722490546965078854542, 9.495741905168625797444037465976, 10.05587395821094842814819089410, 11.00398853278766490273457225152