| L(s) = 1 | + (13.5 − 7.79i)3-s + (77.6 + 44.8i)5-s + (−204. + 275. i)7-s + (121.5 − 210. i)9-s + (1.01e3 + 1.75e3i)11-s − 1.76e3i·13-s + 1.39e3·15-s + (3.38e3 − 1.95e3i)17-s + (1.45e3 + 838. i)19-s + (−609. + 5.31e3i)21-s + (151. − 262. i)23-s + (−3.78e3 − 6.56e3i)25-s − 3.78e3i·27-s + 3.19e4·29-s + (−2.55e4 + 1.47e4i)31-s + ⋯ |
| L(s) = 1 | + (0.5 − 0.288i)3-s + (0.621 + 0.358i)5-s + (−0.595 + 0.803i)7-s + (0.166 − 0.288i)9-s + (0.760 + 1.31i)11-s − 0.803i·13-s + 0.414·15-s + (0.689 − 0.398i)17-s + (0.211 + 0.122i)19-s + (−0.0657 + 0.573i)21-s + (0.0124 − 0.0215i)23-s + (−0.242 − 0.419i)25-s − 0.192i·27-s + 1.31·29-s + (−0.857 + 0.495i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.648376102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.648376102\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-13.5 + 7.79i)T \) |
| 7 | \( 1 + (204. - 275. i)T \) |
| good | 5 | \( 1 + (-77.6 - 44.8i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-1.01e3 - 1.75e3i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + 1.76e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (-3.38e3 + 1.95e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-1.45e3 - 838. i)T + (2.35e7 + 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-151. + 262. i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 - 3.19e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (2.55e4 - 1.47e4i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + (4.43e4 - 7.67e4i)T + (-1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 - 7.22e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.27e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-1.37e4 - 7.93e3i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-1.36e5 - 2.36e5i)T + (-1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (1.37e5 - 7.96e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (3.41e5 + 1.97e5i)T + (2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.36e5 - 2.35e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + 4.95e4T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-3.99e4 + 2.30e4i)T + (7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-3.72e5 + 6.44e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 - 6.58e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (-1.00e6 - 5.80e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 - 6.86e5iT - 8.32e11T^{2} \) |
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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
−10.39998560913681681734132661167, −9.761913498485206798826230229917, −9.014304461560221523604892781989, −7.88140791668545918252090584473, −6.83659194075739683555877014490, −6.05263134217197803612004485839, −4.84220315216352809471101846186, −3.30586560838272770917034463839, −2.44074054026770662567500067967, −1.27960425546007117639387032654,
0.58734024668201646613110959383, 1.76683653057053384164533717451, 3.32348026671491195383363120766, 4.03727988295562631499963762926, 5.47592772387480228132529307108, 6.43818708013730520171012344286, 7.49203153680867026095497105759, 8.751996698247612482430379903914, 9.298482685006640057643708970353, 10.22989096784948730818038263493
