| L(s) = 1 | + (9.69 + 42.4i)2-s + (−28.1 − 26.1i)3-s + (−1.25e3 + 602. i)4-s + (461. − 69.5i)5-s + (837. − 1.45e3i)6-s + (1.97e3 + 3.41e3i)7-s + (−2.37e4 − 2.98e4i)8-s + (−1.36e3 − 1.81e4i)9-s + (7.43e3 + 1.89e4i)10-s + (−7.01e4 − 3.37e4i)11-s + (5.09e4 + 1.57e4i)12-s + (3.02e4 − 7.70e4i)13-s + (−1.26e5 + 1.16e5i)14-s + (−1.48e4 − 1.01e4i)15-s + (5.94e5 − 7.45e5i)16-s + (5.23e5 + 7.88e4i)17-s + ⋯ |
| L(s) = 1 | + (0.428 + 1.87i)2-s + (−0.200 − 0.186i)3-s + (−2.44 + 1.17i)4-s + (0.330 − 0.0497i)5-s + (0.263 − 0.457i)6-s + (0.310 + 0.537i)7-s + (−2.05 − 2.57i)8-s + (−0.0691 − 0.922i)9-s + (0.235 + 0.598i)10-s + (−1.44 − 0.695i)11-s + (0.709 + 0.218i)12-s + (0.293 − 0.748i)13-s + (−0.877 + 0.813i)14-s + (−0.0756 − 0.0515i)15-s + (2.26 − 2.84i)16-s + (1.51 + 0.228i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0124i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.853011 + 0.00531191i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.853011 + 0.00531191i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (2.93e6 + 2.22e7i)T \) |
| good | 2 | \( 1 + (-9.69 - 42.4i)T + (-461. + 222. i)T^{2} \) |
| 3 | \( 1 + (28.1 + 26.1i)T + (1.47e3 + 1.96e4i)T^{2} \) |
| 5 | \( 1 + (-461. + 69.5i)T + (1.86e6 - 5.75e5i)T^{2} \) |
| 7 | \( 1 + (-1.97e3 - 3.41e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (7.01e4 + 3.37e4i)T + (1.47e9 + 1.84e9i)T^{2} \) |
| 13 | \( 1 + (-3.02e4 + 7.70e4i)T + (-7.77e9 - 7.21e9i)T^{2} \) |
| 17 | \( 1 + (-5.23e5 - 7.88e4i)T + (1.13e11 + 3.49e10i)T^{2} \) |
| 19 | \( 1 + (1.33e4 - 1.77e5i)T + (-3.19e11 - 4.80e10i)T^{2} \) |
| 23 | \( 1 + (1.21e6 - 8.27e5i)T + (6.58e11 - 1.67e12i)T^{2} \) |
| 29 | \( 1 + (-3.59e6 + 3.33e6i)T + (1.08e12 - 1.44e13i)T^{2} \) |
| 31 | \( 1 + (4.61e6 + 1.42e6i)T + (2.18e13 + 1.48e13i)T^{2} \) |
| 37 | \( 1 + (-3.78e6 + 6.55e6i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (3.72e6 + 1.63e7i)T + (-2.94e14 + 1.42e14i)T^{2} \) |
| 47 | \( 1 + (2.24e7 - 1.08e7i)T + (6.97e14 - 8.74e14i)T^{2} \) |
| 53 | \( 1 + (-1.05e5 - 2.69e5i)T + (-2.41e15 + 2.24e15i)T^{2} \) |
| 59 | \( 1 + (1.82e7 - 2.28e7i)T + (-1.92e15 - 8.44e15i)T^{2} \) |
| 61 | \( 1 + (-6.63e7 + 2.04e7i)T + (9.66e15 - 6.58e15i)T^{2} \) |
| 67 | \( 1 + (-1.87e7 + 2.49e8i)T + (-2.69e16 - 4.05e15i)T^{2} \) |
| 71 | \( 1 + (6.89e7 + 4.70e7i)T + (1.67e16 + 4.26e16i)T^{2} \) |
| 73 | \( 1 + (1.13e8 - 2.87e8i)T + (-4.31e16 - 4.00e16i)T^{2} \) |
| 79 | \( 1 + (3.54e7 + 6.13e7i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (5.07e8 + 4.70e8i)T + (1.39e16 + 1.86e17i)T^{2} \) |
| 89 | \( 1 + (2.21e8 + 2.05e8i)T + (2.61e16 + 3.49e17i)T^{2} \) |
| 97 | \( 1 + (1.16e9 + 5.58e8i)T + (4.73e17 + 5.94e17i)T^{2} \) |
| 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
−14.20981006918734807200300128139, −13.19353241715671288491417251702, −12.16580351114691761453688087154, −9.881893880863911759295036984095, −8.418458119079476857130179062270, −7.60857145992343015435800316988, −5.79427026799313616104355660937, −5.60119239519378341740723295441, −3.53295157673528284586172116487, −0.27319147802906877042779304160,
1.50050082512043170439709643448, 2.71929447141924153078971128444, 4.40900855778570975371332832885, 5.36300507545704493451348271666, 8.072155904055156697224179408940, 9.891147017515225111927540535389, 10.42341850907120653723200111859, 11.48791367405316848353018285578, 12.67591114100788469710065398314, 13.65904110456157200862370950756
