| L(s) = 1 | + (0.657 + 3.30i)3-s + (−5.77 − 3.85i)5-s + (−10.9 + 7.33i)7-s + (−2.17 + 0.899i)9-s + (−1.56 − 0.311i)11-s + (−3.40 + 3.40i)13-s + (8.94 − 21.5i)15-s + (−16.9 + 1.41i)17-s + (19.7 + 8.16i)19-s + (−31.4 − 31.4i)21-s + (0.0549 − 0.276i)23-s + (8.86 + 21.3i)25-s + (12.4 + 18.6i)27-s + (−14.8 + 22.2i)29-s + (−18.2 + 3.62i)31-s + ⋯ |
| L(s) = 1 | + (0.219 + 1.10i)3-s + (−1.15 − 0.771i)5-s + (−1.56 + 1.04i)7-s + (−0.241 + 0.0998i)9-s + (−0.142 − 0.0283i)11-s + (−0.262 + 0.262i)13-s + (0.596 − 1.43i)15-s + (−0.996 + 0.0832i)17-s + (1.03 + 0.429i)19-s + (−1.49 − 1.49i)21-s + (0.00239 − 0.0120i)23-s + (0.354 + 0.855i)25-s + (0.461 + 0.689i)27-s + (−0.513 + 0.768i)29-s + (−0.588 + 0.117i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.168i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.985 - 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0440063 + 0.517641i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0440063 + 0.517641i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (16.9 - 1.41i)T \) |
| good | 3 | \( 1 + (-0.657 - 3.30i)T + (-8.31 + 3.44i)T^{2} \) |
| 5 | \( 1 + (5.77 + 3.85i)T + (9.56 + 23.0i)T^{2} \) |
| 7 | \( 1 + (10.9 - 7.33i)T + (18.7 - 45.2i)T^{2} \) |
| 11 | \( 1 + (1.56 + 0.311i)T + (111. + 46.3i)T^{2} \) |
| 13 | \( 1 + (3.40 - 3.40i)T - 169iT^{2} \) |
| 19 | \( 1 + (-19.7 - 8.16i)T + (255. + 255. i)T^{2} \) |
| 23 | \( 1 + (-0.0549 + 0.276i)T + (-488. - 202. i)T^{2} \) |
| 29 | \( 1 + (14.8 - 22.2i)T + (-321. - 776. i)T^{2} \) |
| 31 | \( 1 + (18.2 - 3.62i)T + (887. - 367. i)T^{2} \) |
| 37 | \( 1 + (-9.88 - 49.7i)T + (-1.26e3 + 523. i)T^{2} \) |
| 41 | \( 1 + (-51.5 + 34.4i)T + (643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (65.2 - 27.0i)T + (1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-53.3 + 53.3i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (21.3 + 8.85i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-14.2 - 34.3i)T + (-2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-8.93 - 13.3i)T + (-1.42e3 + 3.43e3i)T^{2} \) |
| 67 | \( 1 - 64.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-12.1 - 60.9i)T + (-4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (96.2 + 64.2i)T + (2.03e3 + 4.92e3i)T^{2} \) |
| 79 | \( 1 + (116. + 23.0i)T + (5.76e3 + 2.38e3i)T^{2} \) |
| 83 | \( 1 + (20.2 - 48.8i)T + (-4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-69.5 - 69.5i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (10.6 - 15.9i)T + (-3.60e3 - 8.69e3i)T^{2} \) |
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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
−13.21144765791039883218760398865, −12.39508177966269921463445826126, −11.55745853616718824944799430621, −10.12692292151048023774992032282, −9.246626189670327660424567057468, −8.609364160127335145148270050102, −7.02539543002426481769495865481, −5.45648834568826698663948206050, −4.19045551712253373869928041857, −3.15366307758444007268657303209,
0.32710083438808063924338279994, 2.85741879257275652407815535647, 4.03439434621947362792174200126, 6.36532360544253553087950497607, 7.28790922604401199582629681419, 7.62715534893763372184973460845, 9.383926354383589208830421428159, 10.56613085528973377575995098263, 11.56186575674154773162362530746, 12.75665502155028007352937641906
