L(s) = 1 | + (2.52 − 1.45i)2-s + (8.18 − 4.72i)3-s + (−3.75 + 6.49i)4-s + (−7.00 − 12.1i)5-s + (13.7 − 23.8i)6-s + 19.8·7-s + 68.5i·8-s + (4.16 − 7.20i)9-s + (−35.3 − 20.4i)10-s − 177.·11-s + 70.8i·12-s + (83.2 + 48.0i)13-s + (50.0 − 28.8i)14-s + (−114. − 66.1i)15-s + (39.8 + 69.0i)16-s + (26.8 + 46.5i)17-s + ⋯ |
L(s) = 1 | + (0.631 − 0.364i)2-s + (0.909 − 0.525i)3-s + (−0.234 + 0.406i)4-s + (−0.280 − 0.484i)5-s + (0.382 − 0.662i)6-s + 0.404·7-s + 1.07i·8-s + (0.0513 − 0.0889i)9-s + (−0.353 − 0.204i)10-s − 1.46·11-s + 0.492i·12-s + (0.492 + 0.284i)13-s + (0.255 − 0.147i)14-s + (−0.509 − 0.294i)15-s + (0.155 + 0.269i)16-s + (0.0929 + 0.161i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.76963 - 0.510545i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.76963 - 0.510545i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 19 | \( 1 + (-161. + 322. i)T \) |
good | 2 | \( 1 + (-2.52 + 1.45i)T + (8 - 13.8i)T^{2} \) |
| 3 | \( 1 + (-8.18 + 4.72i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (7.00 + 12.1i)T + (-312.5 + 541. i)T^{2} \) |
| 7 | \( 1 - 19.8T + 2.40e3T^{2} \) |
| 11 | \( 1 + 177.T + 1.46e4T^{2} \) |
| 13 | \( 1 + (-83.2 - 48.0i)T + (1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 + (-26.8 - 46.5i)T + (-4.17e4 + 7.23e4i)T^{2} \) |
| 23 | \( 1 + (-372. + 644. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (738. + 426. i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 - 317. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.69e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-1.74e3 + 1.00e3i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-938. - 1.62e3i)T + (-1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (665. - 1.15e3i)T + (-2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + (2.04e3 + 1.17e3i)T + (3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-2.57e3 + 1.48e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (2.09e3 - 3.63e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-7.66e3 - 4.42e3i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (6.43e3 - 3.71e3i)T + (1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + (4.26e3 + 7.37e3i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-3.80e3 + 2.19e3i)T + (1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 - 1.01e4T + 4.74e7T^{2} \) |
| 89 | \( 1 + (3.23e3 + 1.86e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (-845. + 487. i)T + (4.42e7 - 7.66e7i)T^{2} \) |
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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
−17.83379306671571383697027766574, −16.24312906451703060651092732428, −14.60787303984200502110804256909, −13.42032279550802814107230515243, −12.73842414177145403344232658230, −11.12352568571527838586056676856, −8.686077157419286685006895491616, −7.80167613362777740073198103005, −4.82955360170490391756841062855, −2.72191012136388483351886521091,
3.49749324714296198308031145215, 5.43849830909099737289700156126, 7.69003474028073789197735169257, 9.433260812266362297486602270819, 10.84195034270897078838432729082, 13.04817074040827514070763306529, 14.22809552971273279146031053272, 15.10671036601427590808798215689, 15.89590052831446954150401683739, 18.10459057174320015290546572603