| L(s) = 1 | + (−4.32 − 5.42i)2-s + (83.5 + 12.5i)3-s + (103. − 452. i)4-s + (−697. − 475. i)5-s + (−292. − 507. i)6-s + (3.01e3 − 5.22e3i)7-s + (−6.09e3 + 2.93e3i)8-s + (−1.19e4 − 3.69e3i)9-s + (437. + 5.83e3i)10-s + (−6.45e3 − 2.82e4i)11-s + (1.43e4 − 3.64e4i)12-s + (−1.09e4 + 1.46e5i)13-s + (−4.13e4 + 6.23e3i)14-s + (−5.22e4 − 4.84e4i)15-s + (−1.71e5 − 8.26e4i)16-s + (−5.84e4 + 3.98e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.191 − 0.239i)2-s + (0.595 + 0.0897i)3-s + (0.201 − 0.883i)4-s + (−0.498 − 0.340i)5-s + (−0.0922 − 0.159i)6-s + (0.474 − 0.822i)7-s + (−0.526 + 0.253i)8-s + (−0.609 − 0.187i)9-s + (0.0138 + 0.184i)10-s + (−0.132 − 0.582i)11-s + (0.199 − 0.507i)12-s + (−0.106 + 1.42i)13-s + (−0.287 + 0.0433i)14-s + (−0.266 − 0.247i)15-s + (−0.655 − 0.315i)16-s + (−0.169 + 0.115i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.142i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.0685665 + 0.959989i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0685665 + 0.959989i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.64e6 - 2.23e7i)T \) |
| good | 2 | \( 1 + (4.32 + 5.42i)T + (-113. + 499. i)T^{2} \) |
| 3 | \( 1 + (-83.5 - 12.5i)T + (1.88e4 + 5.80e3i)T^{2} \) |
| 5 | \( 1 + (697. + 475. i)T + (7.13e5 + 1.81e6i)T^{2} \) |
| 7 | \( 1 + (-3.01e3 + 5.22e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (6.45e3 + 2.82e4i)T + (-2.12e9 + 1.02e9i)T^{2} \) |
| 13 | \( 1 + (1.09e4 - 1.46e5i)T + (-1.04e10 - 1.58e9i)T^{2} \) |
| 17 | \( 1 + (5.84e4 - 3.98e4i)T + (4.33e10 - 1.10e11i)T^{2} \) |
| 19 | \( 1 + (-7.97e5 + 2.45e5i)T + (2.66e11 - 1.81e11i)T^{2} \) |
| 23 | \( 1 + (6.44e5 - 5.98e5i)T + (1.34e11 - 1.79e12i)T^{2} \) |
| 29 | \( 1 + (5.34e6 - 8.05e5i)T + (1.38e13 - 4.27e12i)T^{2} \) |
| 31 | \( 1 + (3.04e6 - 7.76e6i)T + (-1.93e13 - 1.79e13i)T^{2} \) |
| 37 | \( 1 + (3.49e6 + 6.04e6i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 + (1.74e7 + 2.18e7i)T + (-7.28e13 + 3.19e14i)T^{2} \) |
| 47 | \( 1 + (-1.37e7 + 6.03e7i)T + (-1.00e15 - 4.85e14i)T^{2} \) |
| 53 | \( 1 + (3.80e6 + 5.08e7i)T + (-3.26e15 + 4.91e14i)T^{2} \) |
| 59 | \( 1 + (-3.01e7 - 1.45e7i)T + (5.40e15 + 6.77e15i)T^{2} \) |
| 61 | \( 1 + (4.66e7 + 1.18e8i)T + (-8.57e15 + 7.95e15i)T^{2} \) |
| 67 | \( 1 + (-2.39e8 + 7.39e7i)T + (2.24e16 - 1.53e16i)T^{2} \) |
| 71 | \( 1 + (-9.76e7 - 9.05e7i)T + (3.42e15 + 4.57e16i)T^{2} \) |
| 73 | \( 1 + (2.51e7 - 3.35e8i)T + (-5.82e16 - 8.77e15i)T^{2} \) |
| 79 | \( 1 + (-2.21e8 + 3.84e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-3.09e8 - 4.67e7i)T + (1.78e17 + 5.51e16i)T^{2} \) |
| 89 | \( 1 + (4.66e8 + 7.03e7i)T + (3.34e17 + 1.03e17i)T^{2} \) |
| 97 | \( 1 + (3.09e8 + 1.35e9i)T + (-6.84e17 + 3.29e17i)T^{2} \) |
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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
−13.88709822402821227737771418839, −11.79217976365738548138316504003, −11.04210217601523311198102625580, −9.574444543438611112623887716428, −8.565140494139051849557902763119, −7.06220749896472332085865980130, −5.30939532038629243662079591899, −3.67217092563984475159399321124, −1.80502965610407075002633453221, −0.31117249658710736374430138074,
2.39846407736043161552208403096, 3.48117448965316241164103439201, 5.56934000360674865892417435352, 7.58502845823078674708873366038, 8.051898644311772562772847598916, 9.387674738804110191894326510418, 11.25185876218860719607579498793, 12.17386708525260946216550257038, 13.38433297235588327746962551034, 14.93136272432197066756920659062
