L(s) = 1 | + (−2 + i)5-s + 4i·7-s − 4·11-s + 4i·13-s + 6i·17-s + 4·19-s + 4i·23-s + (3 − 4i)25-s − 4·29-s + (−4 − 8i)35-s − 4i·37-s + 8·41-s − 12i·47-s − 9·49-s + 2i·53-s + ⋯ |
L(s) = 1 | + (−0.894 + 0.447i)5-s + 1.51i·7-s − 1.20·11-s + 1.10i·13-s + 1.45i·17-s + 0.917·19-s + 0.834i·23-s + (0.600 − 0.800i)25-s − 0.742·29-s + (−0.676 − 1.35i)35-s − 0.657i·37-s + 1.24·41-s − 1.75i·47-s − 1.28·49-s + 0.274i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6543157256\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6543157256\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 - i)T \) |
good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 16iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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
−9.024194744153206254052796956394, −8.487461283048842591902316166351, −7.68960445231581864247947747936, −7.11283471984395898066137506535, −5.96378065035739271085672422017, −5.54583389924879965641187452760, −4.48302779728625038791681746606, −3.57508696283931694946623964982, −2.69508607498131347000657334982, −1.81178749622729775066131397465,
0.24569304183740077259754034444, 1.01009861874939604786914291011, 2.82850980720148956160380265430, 3.43762370323646555213323837738, 4.59963128567168788112929423962, 4.91872394632157759872463135633, 6.00054360703214209432234930621, 7.24312746076055942709068405075, 7.62367486874556955243547348235, 7.978546430325502216157461695627