| 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 | \( 1 - 6.14e7T + 7.83e15T^{2} \) |
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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
−10.42007331019323443847827749659, −8.966646362744282813191804536949, −8.569913536744596429605579071333, −7.54130023342341156471066802183, −6.33375175903154981933851337595, −5.13765918565274460917500358195, −3.99106041799840911053308315903, −3.21122228999452804094125814012, −1.39282532050719499253848801864, −0.17828031718477204054950968028,
0.72344928216618755213107199184, 2.69883197420471856011220902411, 3.72197172223270998831996498045, 4.33446593550488841610179563087, 6.22523467286428518927381577434, 7.05177282517297686263595165286, 7.77855737113232460567301312957, 8.860239589140406684674910185718, 10.17909967218675803026377550190, 10.98764164164868500129531809516
