| L(s) = 1 | + (3.74 − 4.70i)2-s + (29.2 − 4.40i)3-s + (105. + 463. i)4-s + (328. − 224. i)5-s + (88.8 − 153. i)6-s + (2.91e3 + 5.05e3i)7-s + (5.35e3 + 2.57e3i)8-s + (−1.79e4 + 5.54e3i)9-s + (178. − 2.38e3i)10-s + (1.14e4 − 5.03e4i)11-s + (5.13e3 + 1.30e4i)12-s + (1.02e4 + 1.36e5i)13-s + (3.47e4 + 5.23e3i)14-s + (8.62e3 − 8.00e3i)15-s + (−1.87e5 + 9.01e4i)16-s + (−2.45e5 − 1.67e5i)17-s + ⋯ |
| L(s) = 1 | + (0.165 − 0.207i)2-s + (0.208 − 0.0314i)3-s + (0.206 + 0.906i)4-s + (0.235 − 0.160i)5-s + (0.0280 − 0.0484i)6-s + (0.459 + 0.795i)7-s + (0.462 + 0.222i)8-s + (−0.913 + 0.281i)9-s + (0.00565 − 0.0755i)10-s + (0.236 − 1.03i)11-s + (0.0715 + 0.182i)12-s + (0.0994 + 1.32i)13-s + (0.241 + 0.0363i)14-s + (0.0439 − 0.0408i)15-s + (−0.714 + 0.344i)16-s + (−0.713 − 0.486i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.20084 + 1.54380i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20084 + 1.54380i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-2.20e7 + 4.00e6i)T \) |
| good | 2 | \( 1 + (-3.74 + 4.70i)T + (-113. - 499. i)T^{2} \) |
| 3 | \( 1 + (-29.2 + 4.40i)T + (1.88e4 - 5.80e3i)T^{2} \) |
| 5 | \( 1 + (-328. + 224. i)T + (7.13e5 - 1.81e6i)T^{2} \) |
| 7 | \( 1 + (-2.91e3 - 5.05e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-1.14e4 + 5.03e4i)T + (-2.12e9 - 1.02e9i)T^{2} \) |
| 13 | \( 1 + (-1.02e4 - 1.36e5i)T + (-1.04e10 + 1.58e9i)T^{2} \) |
| 17 | \( 1 + (2.45e5 + 1.67e5i)T + (4.33e10 + 1.10e11i)T^{2} \) |
| 19 | \( 1 + (4.76e5 + 1.46e5i)T + (2.66e11 + 1.81e11i)T^{2} \) |
| 23 | \( 1 + (-1.37e6 - 1.27e6i)T + (1.34e11 + 1.79e12i)T^{2} \) |
| 29 | \( 1 + (-2.86e6 - 4.31e5i)T + (1.38e13 + 4.27e12i)T^{2} \) |
| 31 | \( 1 + (-3.97e5 - 1.01e6i)T + (-1.93e13 + 1.79e13i)T^{2} \) |
| 37 | \( 1 + (1.11e7 - 1.93e7i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (1.69e7 - 2.13e7i)T + (-7.28e13 - 3.19e14i)T^{2} \) |
| 47 | \( 1 + (3.75e6 + 1.64e7i)T + (-1.00e15 + 4.85e14i)T^{2} \) |
| 53 | \( 1 + (-2.71e6 + 3.62e7i)T + (-3.26e15 - 4.91e14i)T^{2} \) |
| 59 | \( 1 + (-7.64e7 + 3.68e7i)T + (5.40e15 - 6.77e15i)T^{2} \) |
| 61 | \( 1 + (-3.73e7 + 9.52e7i)T + (-8.57e15 - 7.95e15i)T^{2} \) |
| 67 | \( 1 + (7.60e7 + 2.34e7i)T + (2.24e16 + 1.53e16i)T^{2} \) |
| 71 | \( 1 + (-2.92e8 + 2.71e8i)T + (3.42e15 - 4.57e16i)T^{2} \) |
| 73 | \( 1 + (-1.63e7 - 2.18e8i)T + (-5.82e16 + 8.77e15i)T^{2} \) |
| 79 | \( 1 + (7.89e7 + 1.36e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (7.08e8 - 1.06e8i)T + (1.78e17 - 5.51e16i)T^{2} \) |
| 89 | \( 1 + (-1.60e8 + 2.41e7i)T + (3.34e17 - 1.03e17i)T^{2} \) |
| 97 | \( 1 + (1.03e8 - 4.55e8i)T + (-6.84e17 - 3.29e17i)T^{2} \) |
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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
−14.03513822595000207209764741979, −13.25795755308487011976779278298, −11.64384062634824024653441993064, −11.31897770906897671517796350804, −8.975611862609500114918296075318, −8.408666862843259860484540800501, −6.66807140351432082302092690527, −4.99351248939961959532788756901, −3.24172425953246228143038963443, −1.96886232656644482959545439578,
0.59979419332916720407687548523, 2.27417334342150615272587429688, 4.35937929725734882649989704550, 5.81962019702173384889600874644, 7.07607551476233800456235031260, 8.642761631641955412227397216707, 10.24395690667937790559036116879, 10.88257633694423882916219111983, 12.60300385308581853497568272564, 13.99109595588792600666759619206
