| L(s) = 1 | + (0.642 − 1.26i)2-s + (0.270 − 1.71i)3-s + (−1.17 − 1.61i)4-s + (−3.90 + 3.12i)5-s + (−1.98 − 1.43i)6-s + (−6.00 − 6.00i)7-s + (−2.79 + 0.442i)8-s + (−2.85 − 0.927i)9-s + (1.42 + 6.92i)10-s + (−5.41 − 16.6i)11-s + (−3.08 + 1.57i)12-s + (−0.892 − 1.75i)13-s + (−11.4 + 3.71i)14-s + (4.28 + 7.52i)15-s + (−1.23 + 3.80i)16-s + (2.71 + 17.1i)17-s + ⋯ |
| L(s) = 1 | + (0.321 − 0.630i)2-s + (0.0903 − 0.570i)3-s + (−0.293 − 0.404i)4-s + (−0.781 + 0.624i)5-s + (−0.330 − 0.239i)6-s + (−0.858 − 0.858i)7-s + (−0.349 + 0.0553i)8-s + (−0.317 − 0.103i)9-s + (0.142 + 0.692i)10-s + (−0.491 − 1.51i)11-s + (−0.257 + 0.131i)12-s + (−0.0686 − 0.134i)13-s + (−0.816 + 0.265i)14-s + (0.285 + 0.501i)15-s + (−0.0772 + 0.237i)16-s + (0.159 + 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.153i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0731406 - 0.944318i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0731406 - 0.944318i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.642 + 1.26i)T \) |
| 3 | \( 1 + (-0.270 + 1.71i)T \) |
| 5 | \( 1 + (3.90 - 3.12i)T \) |
| good | 7 | \( 1 + (6.00 + 6.00i)T + 49iT^{2} \) |
| 11 | \( 1 + (5.41 + 16.6i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (0.892 + 1.75i)T + (-99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (-2.71 - 17.1i)T + (-274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (-9.77 + 13.4i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (-22.5 - 11.5i)T + (310. + 427. i)T^{2} \) |
| 29 | \( 1 + (-11.6 - 16.0i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (45.5 + 33.1i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-39.3 + 20.0i)T + (804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-15.1 + 46.6i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (-16.8 + 16.8i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (38.7 + 6.12i)T + (2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (15.0 - 94.8i)T + (-2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (-22.5 - 7.31i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (23.4 + 72.0i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (2.15 + 13.6i)T + (-4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-72.6 + 52.8i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (45.3 + 23.0i)T + (3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-68.3 - 94.0i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (32.7 - 5.18i)T + (6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (44.3 - 14.3i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (36.1 + 5.72i)T + (8.94e3 + 2.90e3i)T^{2} \) |
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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
−12.43468790699201794899918901040, −11.02209090329736527223830024230, −10.84194865699277576639772263437, −9.301267632897459968303115932989, −7.971358298466357994083312477880, −6.95226848233573040655223426304, −5.74513077532018157712249668033, −3.79979174915044505845082480773, −2.97549322776066655485627531103, −0.53990605385004125314048955467,
3.01942191829008782573356158083, 4.51542079795145830259167407320, 5.36051380056697689118383587771, 6.91090511996742103102492554217, 7.982908116806112970840229675716, 9.178450103027058207480221236982, 9.835797491311741351110803100627, 11.52270414175550568745717684134, 12.47249553905086609749730690298, 13.03197967228270279146369581976
