| L(s) = 1 | + (−8.60 + 4.96i)2-s + (−5.90 + 10.2i)3-s + (−14.6 + 25.3i)4-s + (369. + 213. i)5-s − 117. i·6-s + (428. − 741. i)7-s − 1.56e3i·8-s + (1.02e3 + 1.77e3i)9-s − 4.24e3·10-s + 2.17e3·11-s + (−172. − 299. i)12-s + (6.94e3 + 4.01e3i)13-s + 8.51e3i·14-s + (−4.36e3 + 2.52e3i)15-s + (5.88e3 + 1.01e4i)16-s + (−2.81e4 + 1.62e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.760 + 0.439i)2-s + (−0.126 + 0.218i)3-s + (−0.114 + 0.198i)4-s + (1.32 + 0.763i)5-s − 0.221i·6-s + (0.471 − 0.817i)7-s − 1.07i·8-s + (0.468 + 0.810i)9-s − 1.34·10-s + 0.492·11-s + (−0.0288 − 0.0500i)12-s + (0.876 + 0.506i)13-s + 0.828i·14-s + (−0.334 + 0.192i)15-s + (0.359 + 0.622i)16-s + (−1.38 + 0.801i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.786039 + 1.14429i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.786039 + 1.14429i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (2.54e5 - 1.73e5i)T \) |
| good | 2 | \( 1 + (8.60 - 4.96i)T + (64 - 110. i)T^{2} \) |
| 3 | \( 1 + (5.90 - 10.2i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-369. - 213. i)T + (3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-428. + 741. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 - 2.17e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-6.94e3 - 4.01e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (2.81e4 - 1.62e4i)T + (2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-3.15e3 - 1.82e3i)T + (4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 - 5.01e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 2.91e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 2.43e5iT - 2.75e10T^{2} \) |
| 41 | \( 1 + (-7.85e4 + 1.36e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 - 8.55e4iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 6.38e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-4.74e4 - 8.21e4i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (1.53e6 - 8.83e5i)T + (1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.02e6 - 5.91e5i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (8.80e5 - 1.52e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-2.58e6 + 4.47e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + 5.67e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-1.85e6 - 1.07e6i)T + (9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-4.53e6 - 7.85e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-9.77e6 + 5.64e6i)T + (2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + 1.50e7iT - 8.07e13T^{2} \) |
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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
−15.43738600519094128112743293685, −13.82594038320991579995097700627, −13.30553695087980987892066257065, −11.03753667782381399979317401454, −10.14551080935893063824811242907, −8.963032589845524438160996116576, −7.41839698336521472314753685365, −6.26370434199310565592018756300, −4.12365822473040658331435101002, −1.66979574313968894999925850097,
0.894386377547518516846558273737, 2.04565956713208151247565917341, 5.05746879957393856797244312567, 6.29522662825602097920021560833, 8.785268885894674627111757469567, 9.181355323109526418401386090479, 10.54398913167574406179549299627, 11.94471086504324660564601296260, 13.21202695645989606158402988662, 14.34738770151660128902897325066
