| L(s) = 1 | − 18.9·2-s + (27.7 + 47.9i)3-s + 229.·4-s + (−117. − 203. i)5-s + (−524. − 907. i)6-s + (−543. + 941. i)7-s − 1.92e3·8-s + (−441. + 764. i)9-s + (2.22e3 + 3.85e3i)10-s + 2.49e3·11-s + (6.36e3 + 1.10e4i)12-s + (−6.01e3 + 1.04e4i)13-s + (1.02e4 − 1.78e4i)14-s + (6.52e3 − 1.12e4i)15-s + 7.04e3·16-s + (−5.53e3 + 9.59e3i)17-s + ⋯ |
| L(s) = 1 | − 1.67·2-s + (0.592 + 1.02i)3-s + 1.79·4-s + (−0.421 − 0.729i)5-s + (−0.990 − 1.71i)6-s + (−0.599 + 1.03i)7-s − 1.33·8-s + (−0.201 + 0.349i)9-s + (0.704 + 1.21i)10-s + 0.564·11-s + (1.06 + 1.84i)12-s + (−0.759 + 1.31i)13-s + (1.00 − 1.73i)14-s + (0.498 − 0.864i)15-s + 0.430·16-s + (−0.273 + 0.473i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.438 + 0.898i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0358208 - 0.0573693i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0358208 - 0.0573693i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (4.85e5 - 1.90e5i)T \) |
| good | 2 | \( 1 + 18.9T + 128T^{2} \) |
| 3 | \( 1 + (-27.7 - 47.9i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (117. + 203. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (543. - 941. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 - 2.49e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (6.01e3 - 1.04e4i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (5.53e3 - 9.59e3i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.93e4 + 3.34e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (1.93e4 + 3.35e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-1.33e4 + 2.30e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (1.16e5 + 2.02e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (2.26e5 + 3.92e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 - 1.73e5T + 1.94e11T^{2} \) |
| 47 | \( 1 + 2.83e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (3.24e5 + 5.62e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + 1.07e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + (-1.25e6 + 2.17e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-9.52e5 - 1.64e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-4.43e5 + 7.67e5i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (1.73e6 - 2.99e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (2.33e6 - 4.04e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (2.48e6 + 4.31e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (2.69e6 + 4.66e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + 1.39e7T + 8.07e13T^{2} \) |
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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
−14.66252353894839574784537982183, −12.48968099613856786617477452191, −11.28761173023014837284854409372, −9.764457841417254803395589785687, −9.111047130407878019588316208049, −8.513208643101120737964251257075, −6.69798472985339912114187806118, −4.33695059602515636456444965511, −2.25867283387623304475865098643, −0.04583822116941183115137262598,
1.43047448821888413478869996584, 3.12042279807167328347676824711, 6.81780570312347180463847181369, 7.39549809416435730137672575600, 8.362087372831645212571393973554, 9.942168482903312108885350498429, 10.74894114102240438983372311534, 12.30742709089708993522548578012, 13.66053771070970203884937963488, 14.95830584743970712702005205032
