| L(s) = 1 | + (−1.26 + 0.642i)2-s + (−1.71 + 0.270i)3-s + (1.17 − 1.61i)4-s + (−0.888 + 4.92i)5-s + (1.98 − 1.43i)6-s + (−2.71 − 2.71i)7-s + (−0.442 + 2.79i)8-s + (2.85 − 0.927i)9-s + (−2.03 − 6.77i)10-s + (0.549 − 1.69i)11-s + (−1.57 + 3.08i)12-s + (−16.6 − 8.49i)13-s + (5.15 + 1.67i)14-s + (0.187 − 8.65i)15-s + (−1.23 − 3.80i)16-s + (−9.55 − 1.51i)17-s + ⋯ |
| L(s) = 1 | + (−0.630 + 0.321i)2-s + (−0.570 + 0.0903i)3-s + (0.293 − 0.404i)4-s + (−0.177 + 0.984i)5-s + (0.330 − 0.239i)6-s + (−0.387 − 0.387i)7-s + (−0.0553 + 0.349i)8-s + (0.317 − 0.103i)9-s + (−0.203 − 0.677i)10-s + (0.0499 − 0.153i)11-s + (−0.131 + 0.257i)12-s + (−1.28 − 0.653i)13-s + (0.368 + 0.119i)14-s + (0.0125 − 0.577i)15-s + (−0.0772 − 0.237i)16-s + (−0.562 − 0.0890i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.670 + 0.742i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.670 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0377127 - 0.0848673i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0377127 - 0.0848673i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.26 - 0.642i)T \) |
| 3 | \( 1 + (1.71 - 0.270i)T \) |
| 5 | \( 1 + (0.888 - 4.92i)T \) |
| good | 7 | \( 1 + (2.71 + 2.71i)T + 49iT^{2} \) |
| 11 | \( 1 + (-0.549 + 1.69i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (16.6 + 8.49i)T + (99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (9.55 + 1.51i)T + (274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (12.4 + 17.0i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + (9.96 + 19.5i)T + (-310. + 427. i)T^{2} \) |
| 29 | \( 1 + (3.14 - 4.33i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (-18.5 + 13.4i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (18.3 - 35.9i)T + (-804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-14.8 - 45.5i)T + (-1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + (3.79 - 3.79i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-8.65 - 54.6i)T + (-2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (51.4 - 8.15i)T + (2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (44.1 - 14.3i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (16.4 - 50.7i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-110. - 17.5i)T + (4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-11.2 - 8.16i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (59.0 + 115. i)T + (-3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-48.5 + 66.7i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-3.44 + 21.7i)T + (-6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (106. + 34.6i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-27.8 - 175. i)T + (-8.94e3 + 2.90e3i)T^{2} \) |
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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
−12.21125801012793798289027811629, −11.11058734799525909274755668185, −10.38662420317961717729770882805, −9.557003319904163609792183644841, −8.025936179199240946455360853960, −6.97180214456633732913411116484, −6.24034112875333719071211621684, −4.62519909554624337470841689597, −2.72429696789082219534025885217, −0.07347449994782728830550205543,
1.95160301340794552313972169938, 4.13348177110264178090678198183, 5.47664163495662181048077017220, 6.86234779167555641425455523749, 8.063786230778317100662212444635, 9.198725442543717092576704953994, 9.940834768126114804308951742052, 11.22276321972179537828502748576, 12.33937710721463663815201603924, 12.49563893157622465953735639193
