L(s) = 1 | + (−1.00e3 + 583. i)5-s + (−1.69e3 − 1.69e3i)7-s + (2.94e3 + 1.70e3i)11-s + 2.15e4·13-s + (3.99e3 + 2.30e3i)17-s + (−6.74e4 − 1.16e5i)19-s + (−963. + 556. i)23-s + (4.84e5 − 8.39e5i)25-s − 6.12e5i·29-s + (2.80e5 − 4.86e5i)31-s + (2.70e6 + 7.21e5i)35-s + (−1.43e6 − 2.49e6i)37-s + 3.77e6i·41-s + 1.31e6·43-s + (3.88e6 − 2.24e6i)47-s + ⋯ |
L(s) = 1 | + (−1.61 + 0.932i)5-s + (−0.707 − 0.706i)7-s + (0.201 + 0.116i)11-s + 0.755·13-s + (0.0478 + 0.0276i)17-s + (−0.517 − 0.896i)19-s + (−0.00344 + 0.00198i)23-s + (1.24 − 2.14i)25-s − 0.866i·29-s + (0.304 − 0.526i)31-s + (1.80 + 0.481i)35-s + (−0.767 − 1.32i)37-s + 1.33i·41-s + 0.384·43-s + (0.796 − 0.459i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 - 0.957i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.287 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.5237629492\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5237629492\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.69e3 + 1.69e3i)T \) |
good | 5 | \( 1 + (1.00e3 - 583. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-2.94e3 - 1.70e3i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 - 2.15e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + (-3.99e3 - 2.30e3i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (6.74e4 + 1.16e5i)T + (-8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (963. - 556. i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + 6.12e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + (-2.80e5 + 4.86e5i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (1.43e6 + 2.49e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 - 3.77e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 1.31e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-3.88e6 + 2.24e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (1.66e5 + 9.59e4i)T + (3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (8.80e6 + 5.08e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-3.83e6 - 6.63e6i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.64e7 - 2.85e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 3.07e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (1.27e6 - 2.20e6i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (1.67e6 + 2.90e6i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + 2.88e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (8.18e7 - 4.72e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 - 6.14e7T + 7.83e15T^{2} \) |
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
−10.98764164164868500129531809516, −10.17909967218675803026377550190, −8.860239589140406684674910185718, −7.77855737113232460567301312957, −7.05177282517297686263595165286, −6.22523467286428518927381577434, −4.33446593550488841610179563087, −3.72197172223270998831996498045, −2.69883197420471856011220902411, −0.72344928216618755213107199184,
0.17828031718477204054950968028, 1.39282532050719499253848801864, 3.21122228999452804094125814012, 3.99106041799840911053308315903, 5.13765918565274460917500358195, 6.33375175903154981933851337595, 7.54130023342341156471066802183, 8.569913536744596429605579071333, 8.966646362744282813191804536949, 10.42007331019323443847827749659