| L(s) = 1 | + (−1.98 − 0.532i)2-s + (−0.853 + 2.87i)3-s + (0.200 + 0.115i)4-s + (1.13 + 4.86i)5-s + (3.22 − 5.25i)6-s + (−2.37 − 6.58i)7-s + (5.48 + 5.48i)8-s + (−7.54 − 4.91i)9-s + (0.338 − 10.2i)10-s + (−17.6 − 10.2i)11-s + (−0.504 + 0.478i)12-s + (3.04 + 3.04i)13-s + (1.21 + 14.3i)14-s + (−14.9 − 0.895i)15-s + (−8.43 − 14.6i)16-s + (−11.8 + 3.18i)17-s + ⋯ |
| L(s) = 1 | + (−0.993 − 0.266i)2-s + (−0.284 + 0.958i)3-s + (0.0502 + 0.0289i)4-s + (0.226 + 0.973i)5-s + (0.537 − 0.876i)6-s + (−0.339 − 0.940i)7-s + (0.685 + 0.685i)8-s + (−0.837 − 0.545i)9-s + (0.0338 − 1.02i)10-s + (−1.60 − 0.927i)11-s + (−0.0420 + 0.0398i)12-s + (0.233 + 0.233i)13-s + (0.0870 + 1.02i)14-s + (−0.998 − 0.0596i)15-s + (−0.527 − 0.913i)16-s + (−0.698 + 0.187i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 + 0.429i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.903 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.000760314 - 0.00337162i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000760314 - 0.00337162i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.853 - 2.87i)T \) |
| 5 | \( 1 + (-1.13 - 4.86i)T \) |
| 7 | \( 1 + (2.37 + 6.58i)T \) |
| good | 2 | \( 1 + (1.98 + 0.532i)T + (3.46 + 2i)T^{2} \) |
| 11 | \( 1 + (17.6 + 10.2i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-3.04 - 3.04i)T + 169iT^{2} \) |
| 17 | \( 1 + (11.8 - 3.18i)T + (250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (11.1 + 19.3i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (1.52 - 5.69i)T + (-458. - 264.5i)T^{2} \) |
| 29 | \( 1 + 11.9T + 841T^{2} \) |
| 31 | \( 1 + (-30.9 - 17.8i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (4.70 - 17.5i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 - 6.29T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-8.41 + 8.41i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (15.4 - 57.8i)T + (-1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (53.4 - 14.3i)T + (2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (60.6 + 34.9i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (80.4 - 46.4i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (21.9 - 5.88i)T + (3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 27.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (29.0 - 7.78i)T + (4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-110. + 63.8i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-16.3 + 16.3i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (-74.4 + 42.9i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (0.194 - 0.194i)T - 9.40e3iT^{2} \) |
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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
−14.03042663915430818930770290468, −13.36466194881604222665836394451, −11.13474377300775863456339982378, −10.78219417262423571913885102115, −10.09476547421397431983104167992, −9.001293568052506840602217765041, −7.77233465561259091574182414957, −6.24792236107247413901266990382, −4.69115896167910903871264959459, −2.96067196482423789572492904379,
0.00347888048058401288942217919, 2.06873155544767295234714423255, 4.92957060073980032916797702870, 6.20660643609003455474001499538, 7.73269403759213866709204181766, 8.360215829118962953490865215035, 9.421290632406586533436584590296, 10.55680863109134572677659608850, 12.21532850313289367001918089080, 12.84730760237769654762407821948
