L(s) = 1 | + (0.480 + 0.660i)2-s + (1.02 − 3.16i)4-s + (0.615 − 0.847i)5-s + (2.11 − 6.49i)7-s + (5.69 − 1.85i)8-s + 0.855·10-s + (2.57 + 10.6i)11-s + (6.23 − 4.52i)13-s + (5.30 − 1.72i)14-s + (−6.82 − 4.96i)16-s + (−5.52 + 7.59i)17-s + (2.91 + 8.95i)19-s + (−2.05 − 2.82i)20-s + (−5.83 + 6.83i)22-s − 4.82i·23-s + ⋯ |
L(s) = 1 | + (0.240 + 0.330i)2-s + (0.257 − 0.792i)4-s + (0.123 − 0.169i)5-s + (0.301 − 0.927i)7-s + (0.711 − 0.231i)8-s + 0.0855·10-s + (0.233 + 0.972i)11-s + (0.479 − 0.348i)13-s + (0.378 − 0.123i)14-s + (−0.426 − 0.310i)16-s + (−0.324 + 0.447i)17-s + (0.153 + 0.471i)19-s + (−0.102 − 0.141i)20-s + (−0.265 + 0.310i)22-s − 0.209i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.415i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.909 + 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.61931 - 0.352753i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.61931 - 0.352753i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (-2.57 - 10.6i)T \) |
good | 2 | \( 1 + (-0.480 - 0.660i)T + (-1.23 + 3.80i)T^{2} \) |
| 5 | \( 1 + (-0.615 + 0.847i)T + (-7.72 - 23.7i)T^{2} \) |
| 7 | \( 1 + (-2.11 + 6.49i)T + (-39.6 - 28.8i)T^{2} \) |
| 13 | \( 1 + (-6.23 + 4.52i)T + (52.2 - 160. i)T^{2} \) |
| 17 | \( 1 + (5.52 - 7.59i)T + (-89.3 - 274. i)T^{2} \) |
| 19 | \( 1 + (-2.91 - 8.95i)T + (-292. + 212. i)T^{2} \) |
| 23 | \( 1 + 4.82iT - 529T^{2} \) |
| 29 | \( 1 + (44.4 + 14.4i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (15.3 - 11.1i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (14.5 - 44.8i)T + (-1.10e3 - 804. i)T^{2} \) |
| 41 | \( 1 + (-7.22 + 2.34i)T + (1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 - 50.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + (62.9 - 20.4i)T + (1.78e3 - 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-22.5 - 31.0i)T + (-868. + 2.67e3i)T^{2} \) |
| 59 | \( 1 + (-111. - 36.2i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (65.8 + 47.8i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 - 70.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-25.4 + 35.0i)T + (-1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-14.1 + 43.6i)T + (-4.31e3 - 3.13e3i)T^{2} \) |
| 79 | \( 1 + (-69.3 + 50.4i)T + (1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (93.5 - 128. i)T + (-2.12e3 - 6.55e3i)T^{2} \) |
| 89 | \( 1 + 154. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (124. - 90.2i)T + (2.90e3 - 8.94e3i)T^{2} \) |
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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
−13.70499494226725666090452680360, −12.78482729487374850400681656258, −11.25286315138303733638743829067, −10.42120559110195228849606499424, −9.401128095941143134299894726544, −7.73213988468879532303826054251, −6.72157576531602023522189697949, −5.40256558705478333297464704462, −4.11887729480887480595529663965, −1.52738880836153995447445703315,
2.34734929376908244368342398248, 3.78853046455184594520427985215, 5.50426505052499709775746694275, 6.94012814530396042912456758302, 8.319793813669256732939098187987, 9.183701721401080142887965337899, 11.01273709525752925206292297269, 11.52139404530117734951076592350, 12.62681945924802983752740538192, 13.57711667330503357842650092213