| 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 | \( 1 + (1.16e9 - 5.58e8i)T + (4.73e17 - 5.94e17i)T^{2} \) |
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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
−13.65904110456157200862370950756, −12.67591114100788469710065398314, −11.48791367405316848353018285578, −10.42341850907120653723200111859, −9.891147017515225111927540535389, −8.072155904055156697224179408940, −5.36300507545704493451348271666, −4.40900855778570975371332832885, −2.71929447141924153078971128444, −1.50050082512043170439709643448,
0.27319147802906877042779304160, 3.53295157673528284586172116487, 5.60119239519378341740723295441, 5.79427026799313616104355660937, 7.60857145992343015435800316988, 8.418458119079476857130179062270, 9.881893880863911759295036984095, 12.16580351114691761453688087154, 13.19353241715671288491417251702, 14.20981006918734807200300128139
