| L(s) = 1 | + (−4.53 + 9.42i)2-s + (21.7 + 1.62i)3-s + (−28.2 − 35.4i)4-s + (33.0 − 107. i)5-s + (−113. + 197. i)6-s + (−303. + 175. i)7-s + (−190. + 43.4i)8-s + (−250. − 37.7i)9-s + (860. + 798. i)10-s + (−1.23e3 + 1.54e3i)11-s + (−556. − 816. i)12-s + (−1.04e3 + 972. i)13-s + (−273. − 3.65e3i)14-s + (893. − 2.27e3i)15-s + (1.09e3 − 4.81e3i)16-s + (−1.80e3 + 558. i)17-s + ⋯ |
| L(s) = 1 | + (−0.567 + 1.17i)2-s + (0.805 + 0.0603i)3-s + (−0.441 − 0.553i)4-s + (0.264 − 0.857i)5-s + (−0.527 + 0.914i)6-s + (−0.885 + 0.511i)7-s + (−0.371 + 0.0848i)8-s + (−0.343 − 0.0518i)9-s + (0.860 + 0.798i)10-s + (−0.926 + 1.16i)11-s + (−0.322 − 0.472i)12-s + (−0.476 + 0.442i)13-s + (−0.0998 − 1.33i)14-s + (0.264 − 0.674i)15-s + (0.268 − 1.17i)16-s + (−0.368 + 0.113i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.177433 - 0.550432i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.177433 - 0.550432i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (6.17e4 + 5.00e4i)T \) |
| good | 2 | \( 1 + (4.53 - 9.42i)T + (-39.9 - 50.0i)T^{2} \) |
| 3 | \( 1 + (-21.7 - 1.62i)T + (720. + 108. i)T^{2} \) |
| 5 | \( 1 + (-33.0 + 107. i)T + (-1.29e4 - 8.80e3i)T^{2} \) |
| 7 | \( 1 + (303. - 175. i)T + (5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (1.23e3 - 1.54e3i)T + (-3.94e5 - 1.72e6i)T^{2} \) |
| 13 | \( 1 + (1.04e3 - 972. i)T + (3.60e5 - 4.81e6i)T^{2} \) |
| 17 | \( 1 + (1.80e3 - 558. i)T + (1.99e7 - 1.35e7i)T^{2} \) |
| 19 | \( 1 + (151. + 1.00e3i)T + (-4.49e7 + 1.38e7i)T^{2} \) |
| 23 | \( 1 + (59.8 + 152. i)T + (-1.08e8 + 1.00e8i)T^{2} \) |
| 29 | \( 1 + (-3.68e4 + 2.76e3i)T + (5.88e8 - 8.86e7i)T^{2} \) |
| 31 | \( 1 + (2.57e4 - 1.75e4i)T + (3.24e8 - 8.26e8i)T^{2} \) |
| 37 | \( 1 + (-7.69e4 - 4.44e4i)T + (1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-9.64e3 - 4.64e3i)T + (2.96e9 + 3.71e9i)T^{2} \) |
| 47 | \( 1 + (4.83e4 + 6.06e4i)T + (-2.39e9 + 1.05e10i)T^{2} \) |
| 53 | \( 1 + (7.38e4 + 6.85e4i)T + (1.65e9 + 2.21e10i)T^{2} \) |
| 59 | \( 1 + (-3.88e4 + 1.70e5i)T + (-3.80e10 - 1.83e10i)T^{2} \) |
| 61 | \( 1 + (2.08e5 - 3.05e5i)T + (-1.88e10 - 4.79e10i)T^{2} \) |
| 67 | \( 1 + (-813. + 122. i)T + (8.64e10 - 2.66e10i)T^{2} \) |
| 71 | \( 1 + (6.78e4 + 2.66e4i)T + (9.39e10 + 8.71e10i)T^{2} \) |
| 73 | \( 1 + (-3.33e5 - 3.59e5i)T + (-1.13e10 + 1.50e11i)T^{2} \) |
| 79 | \( 1 + (-1.93e5 - 3.35e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (1.63e3 - 2.18e4i)T + (-3.23e11 - 4.87e10i)T^{2} \) |
| 89 | \( 1 + (-8.56e5 - 6.42e4i)T + (4.91e11 + 7.40e10i)T^{2} \) |
| 97 | \( 1 + (4.33e5 - 5.43e5i)T + (-1.85e11 - 8.12e11i)T^{2} \) |
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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
−15.49158009084701016120605809389, −14.66248772668182536157023666161, −13.21728226958997176120344124927, −12.17389314835221869776914759140, −9.767633447934532380010045834083, −9.013117594946193409279893000298, −8.025451474526634011397054903147, −6.63206912911295878344954010711, −5.10608416916363031825212691230, −2.63811785130182412773199370724,
0.27114973035534011096948715738, 2.60719810221623661780529617110, 3.22752954340568764775190975669, 6.20455571570481290694119004221, 7.993428475412829511523145092293, 9.333163351622460858015516209285, 10.39287260333287385524280441441, 11.16403480100466176312154150992, 12.81688118903595699162351704707, 13.79629722148218489917035984315
