| L(s) = 1 | + (−0.913 − 1.14i)2-s + (31.0 + 4.67i)3-s + (28.0 − 122. i)4-s + (−325. − 222. i)5-s + (−23.0 − 39.8i)6-s + (−420. + 727. i)7-s + (−335. + 161. i)8-s + (−1.14e3 − 354. i)9-s + (43.1 + 576. i)10-s + (1.28e3 + 5.61e3i)11-s + (1.44e3 − 3.67e3i)12-s + (−549. + 7.33e3i)13-s + (1.21e3 − 183. i)14-s + (−9.06e3 − 8.41e3i)15-s + (−1.40e4 − 6.75e3i)16-s + (2.34e4 − 1.60e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.0807 − 0.101i)2-s + (0.663 + 0.100i)3-s + (0.218 − 0.958i)4-s + (−1.16 − 0.794i)5-s + (−0.0434 − 0.0753i)6-s + (−0.463 + 0.801i)7-s + (−0.231 + 0.111i)8-s + (−0.525 − 0.161i)9-s + (0.0136 + 0.182i)10-s + (0.290 + 1.27i)11-s + (0.241 − 0.614i)12-s + (−0.0694 + 0.926i)13-s + (0.118 − 0.0178i)14-s + (−0.693 − 0.643i)15-s + (−0.855 − 0.412i)16-s + (1.15 − 0.790i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.00871920 + 0.0300629i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00871920 + 0.0300629i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-2.47e5 + 4.58e5i)T \) |
| good | 2 | \( 1 + (0.913 + 1.14i)T + (-28.4 + 124. i)T^{2} \) |
| 3 | \( 1 + (-31.0 - 4.67i)T + (2.08e3 + 644. i)T^{2} \) |
| 5 | \( 1 + (325. + 222. i)T + (2.85e4 + 7.27e4i)T^{2} \) |
| 7 | \( 1 + (420. - 727. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-1.28e3 - 5.61e3i)T + (-1.75e7 + 8.45e6i)T^{2} \) |
| 13 | \( 1 + (549. - 7.33e3i)T + (-6.20e7 - 9.35e6i)T^{2} \) |
| 17 | \( 1 + (-2.34e4 + 1.60e4i)T + (1.49e8 - 3.81e8i)T^{2} \) |
| 19 | \( 1 + (2.34e4 - 7.24e3i)T + (7.38e8 - 5.03e8i)T^{2} \) |
| 23 | \( 1 + (6.17e4 - 5.72e4i)T + (2.54e8 - 3.39e9i)T^{2} \) |
| 29 | \( 1 + (1.77e5 - 2.67e4i)T + (1.64e10 - 5.08e9i)T^{2} \) |
| 31 | \( 1 + (-1.14e5 + 2.91e5i)T + (-2.01e10 - 1.87e10i)T^{2} \) |
| 37 | \( 1 + (1.10e5 + 1.90e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (4.97e5 + 6.24e5i)T + (-4.33e10 + 1.89e11i)T^{2} \) |
| 47 | \( 1 + (6.03e4 - 2.64e5i)T + (-4.56e11 - 2.19e11i)T^{2} \) |
| 53 | \( 1 + (-6.94e4 - 9.26e5i)T + (-1.16e12 + 1.75e11i)T^{2} \) |
| 59 | \( 1 + (2.25e6 + 1.08e6i)T + (1.55e12 + 1.94e12i)T^{2} \) |
| 61 | \( 1 + (-1.16e6 - 2.96e6i)T + (-2.30e12 + 2.13e12i)T^{2} \) |
| 67 | \( 1 + (-8.74e5 + 2.69e5i)T + (5.00e12 - 3.41e12i)T^{2} \) |
| 71 | \( 1 + (-1.38e6 - 1.28e6i)T + (6.79e11 + 9.06e12i)T^{2} \) |
| 73 | \( 1 + (-9.57e4 + 1.27e6i)T + (-1.09e13 - 1.64e12i)T^{2} \) |
| 79 | \( 1 + (6.54e4 - 1.13e5i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (3.39e6 + 5.11e5i)T + (2.59e13 + 7.99e12i)T^{2} \) |
| 89 | \( 1 + (-4.41e5 - 6.64e4i)T + (4.22e13 + 1.30e13i)T^{2} \) |
| 97 | \( 1 + (-5.14e5 - 2.25e6i)T + (-7.27e13 + 3.50e13i)T^{2} \) |
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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
−14.03578878647847698835291846289, −12.23708353707280385926946845509, −11.65807703799090680917751016233, −9.693062766556379218196645908965, −8.991243446360212392728924194093, −7.50223738541017602660136442239, −5.68040824798488443771262256630, −4.04257352072491747725282345837, −2.06063362323100786299432525078, −0.01164590331306332262987573406,
3.12692011171643464045612749447, 3.63427592594152727768556026412, 6.51823723102773916502104857660, 7.920693050546986265025594228389, 8.342996354414015798395182058525, 10.50934620586257525911273061569, 11.55146262621899729370662559690, 12.81583887491498368994379412182, 14.05289717037873877619910108112, 15.09635939930264142011839151460
