| L(s) = 1 | + (19.4 + 33.7i)2-s + (89.2 + 154. i)3-s + (−503. + 871. i)4-s + (−249. − 431. i)5-s + (−3.47e3 + 6.02e3i)6-s + 698.·7-s − 1.92e4·8-s + (−6.10e3 + 1.05e4i)9-s + (9.71e3 − 1.68e4i)10-s + 7.30e4·11-s − 1.79e5·12-s + (−1.00e4 + 1.74e4i)13-s + (1.36e4 + 2.35e4i)14-s + (4.45e4 − 7.71e4i)15-s + (−1.17e5 − 2.03e5i)16-s + (−2.80e5 − 4.85e5i)17-s + ⋯ |
| L(s) = 1 | + (0.861 + 1.49i)2-s + (0.636 + 1.10i)3-s + (−0.982 + 1.70i)4-s + (−0.178 − 0.308i)5-s + (−1.09 + 1.89i)6-s + 0.109·7-s − 1.66·8-s + (−0.309 + 0.536i)9-s + (0.307 − 0.532i)10-s + 1.50·11-s − 2.50·12-s + (−0.0979 + 0.169i)13-s + (0.0946 + 0.163i)14-s + (0.227 − 0.393i)15-s + (−0.449 − 0.777i)16-s + (−0.813 − 1.40i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00726i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.0111598 - 3.07357i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0111598 - 3.07357i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (1.69e5 - 5.42e5i)T \) |
| good | 2 | \( 1 + (-19.4 - 33.7i)T + (-256 + 443. i)T^{2} \) |
| 3 | \( 1 + (-89.2 - 154. i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (249. + 431. i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 - 698.T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.30e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (1.00e4 - 1.74e4i)T + (-5.30e9 - 9.18e9i)T^{2} \) |
| 17 | \( 1 + (2.80e5 + 4.85e5i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 23 | \( 1 + (9.19e5 - 1.59e6i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (-4.54e5 + 7.87e5i)T + (-7.25e12 - 1.25e13i)T^{2} \) |
| 31 | \( 1 - 7.56e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 9.38e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (5.10e5 + 8.84e5i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 + (-1.05e7 - 1.83e7i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 + (-9.01e6 + 1.56e7i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-4.18e7 + 7.24e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-5.02e6 - 8.69e6i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-7.56e7 + 1.31e8i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-8.89e7 + 1.53e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (-2.76e7 - 4.79e7i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 + (1.49e8 + 2.58e8i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-1.23e8 - 2.13e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 - 1.32e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-1.90e8 + 3.30e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + (3.30e8 + 5.72e8i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
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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
−16.36693569132932304932656649821, −15.65979072102674318290453366436, −14.54395119434994993011094048521, −13.82552054600810471079462948710, −11.97043998178913997756393517413, −9.537861619819200474236658034821, −8.343889777930348891986636936353, −6.60111493829429194098584231267, −4.75267587621826824278063319731, −3.75680233197239306740261336143,
1.23541857368529750906744802125, 2.50722707209467390200686977097, 4.14939310775236930687646489887, 6.63079329155590996175934074294, 8.727748875815015361589531915193, 10.62149684680927287903809181869, 11.93874902027109032833397633395, 12.90081982511208472010447470943, 13.94373713159534683959476816173, 14.87947139257911953786251325803
