| L(s) = 1 | + (−2.31 − 7.12i)2-s + (−16.2 + 14.6i)3-s + (6.33 − 4.60i)4-s + (−92.6 − 160. i)5-s + (142. + 82.1i)6-s + (60.8 + 578. i)7-s + (−435. − 316. i)8-s + (−25.9 + 246. i)9-s + (−929. + 1.03e3i)10-s + (65.5 − 147. i)11-s + (−35.6 + 167. i)12-s + (774. + 3.64e3i)13-s + (3.98e3 − 1.77e3i)14-s + (3.86e3 + 1.25e3i)15-s + (−1.09e3 + 3.36e3i)16-s + (−21.4 − 48.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.289 − 0.890i)2-s + (−0.603 + 0.543i)3-s + (0.0989 − 0.0719i)4-s + (−0.741 − 1.28i)5-s + (0.658 + 0.380i)6-s + (0.177 + 1.68i)7-s + (−0.850 − 0.618i)8-s + (−0.0355 + 0.338i)9-s + (−0.929 + 1.03i)10-s + (0.0492 − 0.110i)11-s + (−0.0206 + 0.0971i)12-s + (0.352 + 1.65i)13-s + (1.45 − 0.646i)14-s + (1.14 + 0.372i)15-s + (−0.266 + 0.820i)16-s + (−0.00437 − 0.00981i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0568 - 0.998i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0568 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.292050 + 0.275899i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.292050 + 0.275899i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (-4.20e3 + 2.94e4i)T \) |
| good | 2 | \( 1 + (2.31 + 7.12i)T + (-51.7 + 37.6i)T^{2} \) |
| 3 | \( 1 + (16.2 - 14.6i)T + (76.2 - 725. i)T^{2} \) |
| 5 | \( 1 + (92.6 + 160. i)T + (-7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + (-60.8 - 578. i)T + (-1.15e5 + 2.44e4i)T^{2} \) |
| 11 | \( 1 + (-65.5 + 147. i)T + (-1.18e6 - 1.31e6i)T^{2} \) |
| 13 | \( 1 + (-774. - 3.64e3i)T + (-4.40e6 + 1.96e6i)T^{2} \) |
| 17 | \( 1 + (21.4 + 48.2i)T + (-1.61e7 + 1.79e7i)T^{2} \) |
| 19 | \( 1 + (7.81e3 + 1.66e3i)T + (4.29e7 + 1.91e7i)T^{2} \) |
| 23 | \( 1 + (2.11e3 - 2.91e3i)T + (-4.57e7 - 1.40e8i)T^{2} \) |
| 29 | \( 1 + (3.18e4 - 1.03e4i)T + (4.81e8 - 3.49e8i)T^{2} \) |
| 37 | \( 1 + (-5.88e4 - 3.39e4i)T + (1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (4.65e4 - 5.16e4i)T + (-4.96e8 - 4.72e9i)T^{2} \) |
| 43 | \( 1 + (-2.45e4 + 1.15e5i)T + (-5.77e9 - 2.57e9i)T^{2} \) |
| 47 | \( 1 + (3.56e4 - 1.09e5i)T + (-8.72e9 - 6.33e9i)T^{2} \) |
| 53 | \( 1 + (6.68e4 + 7.03e3i)T + (2.16e10 + 4.60e9i)T^{2} \) |
| 59 | \( 1 + (2.21e3 + 2.46e3i)T + (-4.40e9 + 4.19e10i)T^{2} \) |
| 61 | \( 1 - 9.37e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 + (6.79e4 + 1.17e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + (4.63e4 - 4.41e5i)T + (-1.25e11 - 2.66e10i)T^{2} \) |
| 73 | \( 1 + (6.87e4 - 1.54e5i)T + (-1.01e11 - 1.12e11i)T^{2} \) |
| 79 | \( 1 + (9.72e4 + 2.18e5i)T + (-1.62e11 + 1.80e11i)T^{2} \) |
| 83 | \( 1 + (1.88e5 + 1.70e5i)T + (3.41e10 + 3.25e11i)T^{2} \) |
| 89 | \( 1 + (1.80e5 + 2.48e5i)T + (-1.53e11 + 4.72e11i)T^{2} \) |
| 97 | \( 1 + (1.13e4 - 8.23e3i)T + (2.57e11 - 7.92e11i)T^{2} \) |
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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
−15.99959100301508631953146205304, −15.11446695089239332111198376274, −12.83239684311920318858546920228, −11.71282336350936631333498314964, −11.32243975109829099579771345659, −9.442542449118115807159567353372, −8.584401847143753961496703838370, −5.90570068555864619164612035689, −4.42763218185490404397524419717, −1.93737973246518249375766300096,
0.24416267523530146688398154149, 3.51115443158531249453922591210, 6.21724498750117467207399358153, 7.18293057400619808313277890631, 7.925005046688594724630987792945, 10.52499604769630818718845199467, 11.32080512365023351864584953449, 12.84258886216964630819090310621, 14.54189744526227446435395452791, 15.30171338461040994003681118007
