| L(s) = 1 | + (−9.69 + 7.04i)2-s + (1.23 − 3.79i)3-s + (4.83 − 14.8i)4-s + (70.6 − 270. i)5-s + (14.7 + 45.5i)6-s + 219.·7-s + (−416. − 1.28e3i)8-s + (1.75e3 + 1.27e3i)9-s + (1.22e3 + 3.11e3i)10-s + (−2.27e3 + 1.65e3i)11-s + (−50.5 − 36.7i)12-s + (1.22e4 + 8.91e3i)13-s + (−2.13e3 + 1.54e3i)14-s + (−940. − 602. i)15-s + (1.46e4 + 1.06e4i)16-s + (8.44e3 + 2.59e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.857 + 0.622i)2-s + (0.0263 − 0.0812i)3-s + (0.0377 − 0.116i)4-s + (0.252 − 0.967i)5-s + (0.0279 + 0.0860i)6-s + 0.242·7-s + (−0.287 − 0.884i)8-s + (0.803 + 0.583i)9-s + (0.385 + 0.986i)10-s + (−0.514 + 0.373i)11-s + (−0.00844 − 0.00613i)12-s + (1.54 + 1.12i)13-s + (−0.207 + 0.150i)14-s + (−0.0719 − 0.0460i)15-s + (0.895 + 0.650i)16-s + (0.416 + 1.28i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.06018 + 0.493666i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06018 + 0.493666i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-70.6 + 270. i)T \) |
| good | 2 | \( 1 + (9.69 - 7.04i)T + (39.5 - 121. i)T^{2} \) |
| 3 | \( 1 + (-1.23 + 3.79i)T + (-1.76e3 - 1.28e3i)T^{2} \) |
| 7 | \( 1 - 219.T + 8.23e5T^{2} \) |
| 11 | \( 1 + (2.27e3 - 1.65e3i)T + (6.02e6 - 1.85e7i)T^{2} \) |
| 13 | \( 1 + (-1.22e4 - 8.91e3i)T + (1.93e7 + 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-8.44e3 - 2.59e4i)T + (-3.31e8 + 2.41e8i)T^{2} \) |
| 19 | \( 1 + (9.41e3 + 2.89e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + (-8.31e4 + 6.03e4i)T + (1.05e9 - 3.23e9i)T^{2} \) |
| 29 | \( 1 + (-2.77e4 + 8.54e4i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-8.59e4 - 2.64e5i)T + (-2.22e10 + 1.61e10i)T^{2} \) |
| 37 | \( 1 + (-2.08e5 - 1.51e5i)T + (2.93e10 + 9.02e10i)T^{2} \) |
| 41 | \( 1 + (9.79e4 + 7.11e4i)T + (6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 - 4.96e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-2.21e5 + 6.81e5i)T + (-4.09e11 - 2.97e11i)T^{2} \) |
| 53 | \( 1 + (2.23e5 - 6.88e5i)T + (-9.50e11 - 6.90e11i)T^{2} \) |
| 59 | \( 1 + (5.56e5 + 4.04e5i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 + (8.50e5 - 6.18e5i)T + (9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 + (9.36e5 + 2.88e6i)T + (-4.90e12 + 3.56e12i)T^{2} \) |
| 71 | \( 1 + (-6.51e5 + 2.00e6i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 + (2.25e6 - 1.63e6i)T + (3.41e12 - 1.05e13i)T^{2} \) |
| 79 | \( 1 + (7.21e5 - 2.22e6i)T + (-1.55e13 - 1.12e13i)T^{2} \) |
| 83 | \( 1 + (-9.51e5 - 2.92e6i)T + (-2.19e13 + 1.59e13i)T^{2} \) |
| 89 | \( 1 + (-2.60e6 + 1.89e6i)T + (1.36e13 - 4.20e13i)T^{2} \) |
| 97 | \( 1 + (6.93e5 - 2.13e6i)T + (-6.53e13 - 4.74e13i)T^{2} \) |
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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
−16.36769152745891273228867544055, −15.47104392493049753970412312181, −13.46027774217499241135035227640, −12.58332275151525013216972189580, −10.57994898945543078580419927615, −9.038496641365135941681528341094, −8.173753664730580105665474757104, −6.59028151137464284674166005580, −4.47111752552905725571916976960, −1.28127749319742877166074867940,
1.06139672808095788595448514077, 3.10589123361097108697980231931, 5.82272326155103278307087335800, 7.76878016756612034641990030781, 9.406271149650517981518100434800, 10.50718108581321291177474379097, 11.36429342424569387718030669932, 13.25734200515065640517376824457, 14.66136217962937648138455893687, 15.79448705924505793917928097409
