| L(s) = 1 | + (−12.9 − 9.37i)2-s + 24.3·3-s + (39.0 + 120. i)4-s + (51.9 + 160. i)5-s + (−313. − 227. i)6-s + (338. − 246. i)7-s + (−8.60 + 26.4i)8-s − 1.59e3·9-s + (828. − 2.55e3i)10-s + (2.37e3 − 7.30e3i)11-s + (948. + 2.91e3i)12-s + (−6.22e3 − 4.52e3i)13-s − 6.67e3·14-s + (1.26e3 + 3.89e3i)15-s + (1.34e4 − 9.75e3i)16-s + (−7.94e3 + 2.44e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.14 − 0.828i)2-s + 0.519·3-s + (0.304 + 0.938i)4-s + (0.185 + 0.572i)5-s + (−0.592 − 0.430i)6-s + (0.373 − 0.271i)7-s + (−0.00594 + 0.0182i)8-s − 0.729·9-s + (0.262 − 0.806i)10-s + (0.537 − 1.65i)11-s + (0.158 + 0.487i)12-s + (−0.785 − 0.570i)13-s − 0.650·14-s + (0.0966 + 0.297i)15-s + (0.819 − 0.595i)16-s + (−0.392 + 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.546i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0576421 + 0.193889i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0576421 + 0.193889i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 41 | \( 1 + (-1.28e5 + 4.22e5i)T \) |
| good | 2 | \( 1 + (12.9 + 9.37i)T + (39.5 + 121. i)T^{2} \) |
| 3 | \( 1 - 24.3T + 2.18e3T^{2} \) |
| 5 | \( 1 + (-51.9 - 160. i)T + (-6.32e4 + 4.59e4i)T^{2} \) |
| 7 | \( 1 + (-338. + 246. i)T + (2.54e5 - 7.83e5i)T^{2} \) |
| 11 | \( 1 + (-2.37e3 + 7.30e3i)T + (-1.57e7 - 1.14e7i)T^{2} \) |
| 13 | \( 1 + (6.22e3 + 4.52e3i)T + (1.93e7 + 5.96e7i)T^{2} \) |
| 17 | \( 1 + (7.94e3 - 2.44e4i)T + (-3.31e8 - 2.41e8i)T^{2} \) |
| 19 | \( 1 + (4.50e4 - 3.27e4i)T + (2.76e8 - 8.50e8i)T^{2} \) |
| 23 | \( 1 + (5.00e4 + 3.63e4i)T + (1.05e9 + 3.23e9i)T^{2} \) |
| 29 | \( 1 + (1.10e4 + 3.40e4i)T + (-1.39e10 + 1.01e10i)T^{2} \) |
| 31 | \( 1 + (7.91e4 - 2.43e5i)T + (-2.22e10 - 1.61e10i)T^{2} \) |
| 37 | \( 1 + (2.54e4 + 7.82e4i)T + (-7.68e10 + 5.57e10i)T^{2} \) |
| 43 | \( 1 + (4.20e5 + 3.05e5i)T + (8.39e10 + 2.58e11i)T^{2} \) |
| 47 | \( 1 + (7.21e5 + 5.24e5i)T + (1.56e11 + 4.81e11i)T^{2} \) |
| 53 | \( 1 + (-2.35e5 - 7.24e5i)T + (-9.50e11 + 6.90e11i)T^{2} \) |
| 59 | \( 1 + (2.14e6 + 1.55e6i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 + (5.36e5 - 3.90e5i)T + (9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 + (-1.12e6 - 3.46e6i)T + (-4.90e12 + 3.56e12i)T^{2} \) |
| 71 | \( 1 + (-1.35e6 + 4.15e6i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 - 1.68e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.16e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.75e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-3.80e6 + 2.76e6i)T + (1.36e13 - 4.20e13i)T^{2} \) |
| 97 | \( 1 + (-3.06e5 - 9.43e5i)T + (-6.53e13 + 4.74e13i)T^{2} \) |
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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
−14.14627528962224353566410411588, −12.32643856941480758692726627484, −10.92726959360627806814385983869, −10.39720342191056655706583944852, −8.710801632528396980869651403211, −8.192714806252914597872014468747, −6.06808412830210832989200723538, −3.38255991865535481845152066439, −1.99231055306961490181220192400, −0.10964435719027602368798308463,
2.04060390772027242396450641581, 4.72971712589037858305609047555, 6.68428114639725071659723384028, 7.85526990243461283144952577239, 9.116676049986731437872694064277, 9.575163288762942680846171700300, 11.58548954368361428522094632497, 12.98510873614140347858310529026, 14.62742056861562126979428988102, 15.25615225617942932350764446754
