L(s) = 1 | + (−21.2 − 21.2i)3-s + (−17.6 + 17.6i)5-s + 169. i·7-s + 660. i·9-s + (−379. + 379. i)11-s + (10.8 + 10.8i)13-s + 751.·15-s − 1.78e3·17-s + (1.09e3 + 1.09e3i)19-s + (3.60e3 − 3.60e3i)21-s + 2.09e3i·23-s − 625i·25-s + (8.86e3 − 8.86e3i)27-s + (100. + 100. i)29-s − 3.15e3·31-s + ⋯ |
L(s) = 1 | + (−1.36 − 1.36i)3-s + (−0.316 + 0.316i)5-s + 1.30i·7-s + 2.71i·9-s + (−0.944 + 0.944i)11-s + (0.0177 + 0.0177i)13-s + 0.862·15-s − 1.50·17-s + (0.694 + 0.694i)19-s + (1.78 − 1.78i)21-s + 0.826i·23-s − 0.200i·25-s + (2.34 − 2.34i)27-s + (0.0221 + 0.0221i)29-s − 0.589·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.09644201387\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09644201387\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (17.6 - 17.6i)T \) |
good | 3 | \( 1 + (21.2 + 21.2i)T + 243iT^{2} \) |
| 7 | \( 1 - 169. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (379. - 379. i)T - 1.61e5iT^{2} \) |
| 13 | \( 1 + (-10.8 - 10.8i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + 1.78e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + (-1.09e3 - 1.09e3i)T + 2.47e6iT^{2} \) |
| 23 | \( 1 - 2.09e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (-100. - 100. i)T + 2.05e7iT^{2} \) |
| 31 | \( 1 + 3.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-7.28e3 + 7.28e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 - 1.30e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (1.35e3 - 1.35e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + 1.84e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + (1.01e4 - 1.01e4i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 + (5.69e3 - 5.69e3i)T - 7.14e8iT^{2} \) |
| 61 | \( 1 + (1.68e4 + 1.68e4i)T + 8.44e8iT^{2} \) |
| 67 | \( 1 + (8.86e3 + 8.86e3i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 - 2.31e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 134. iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 2.69e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-2.02e4 - 2.02e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 - 1.01e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 4.78e4T + 8.58e9T^{2} \) |
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
−11.45266688283694889002763488074, −10.87787699164344508613290381190, −9.573315207953826467118095716859, −8.124192783278745944957359888618, −7.42149786111252832041443840485, −6.46175795871044656148838862777, −5.62618798403695028473574263560, −4.78848623922744374938810259620, −2.55241555718147493808055228013, −1.67374881258008566196313711949,
0.04982634003135695414663447961, 0.66358457011250492476929374278, 3.31612070284328308906881372917, 4.38074985189316419111922875682, 4.96745436340452371317189167275, 6.12009061671007823225453553144, 7.11652253615012551290112268278, 8.548335039187956351991667832089, 9.582117254199453978663018920192, 10.57242120999478914054027081246