| L(s) = 1 | + (−0.131 − 0.254i)2-s + (0.487 − 1.66i)3-s + (2.27 − 3.19i)4-s + (4.91 + 4.25i)5-s + (−0.487 + 0.0939i)6-s + (9.56 + 0.455i)7-s + (−2.24 − 0.322i)8-s + (−2.52 − 1.62i)9-s + (0.439 − 1.81i)10-s + (−2.94 + 15.2i)11-s + (−4.19 − 5.33i)12-s + (−1.26 + 3.15i)13-s + (−1.13 − 2.49i)14-s + (9.47 − 6.09i)15-s + (−4.91 − 14.1i)16-s + (−3.03 − 4.26i)17-s + ⋯ |
| L(s) = 1 | + (−0.0656 − 0.127i)2-s + (0.162 − 0.553i)3-s + (0.568 − 0.797i)4-s + (0.983 + 0.851i)5-s + (−0.0812 + 0.0156i)6-s + (1.36 + 0.0650i)7-s + (−0.280 − 0.0403i)8-s + (−0.280 − 0.180i)9-s + (0.0439 − 0.181i)10-s + (−0.267 + 1.38i)11-s + (−0.349 − 0.444i)12-s + (−0.0972 + 0.242i)13-s + (−0.0814 − 0.178i)14-s + (0.631 − 0.406i)15-s + (−0.307 − 0.887i)16-s + (−0.178 − 0.251i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.07153 - 0.640814i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.07153 - 0.640814i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.487 + 1.66i)T \) |
| 67 | \( 1 + (63.8 - 20.3i)T \) |
| good | 2 | \( 1 + (0.131 + 0.254i)T + (-2.32 + 3.25i)T^{2} \) |
| 5 | \( 1 + (-4.91 - 4.25i)T + (3.55 + 24.7i)T^{2} \) |
| 7 | \( 1 + (-9.56 - 0.455i)T + (48.7 + 4.65i)T^{2} \) |
| 11 | \( 1 + (2.94 - 15.2i)T + (-112. - 44.9i)T^{2} \) |
| 13 | \( 1 + (1.26 - 3.15i)T + (-122. - 116. i)T^{2} \) |
| 17 | \( 1 + (3.03 + 4.26i)T + (-94.5 + 273. i)T^{2} \) |
| 19 | \( 1 + (-0.0760 - 1.59i)T + (-359. + 34.3i)T^{2} \) |
| 23 | \( 1 + (-4.23 + 4.03i)T + (25.1 - 528. i)T^{2} \) |
| 29 | \( 1 + (21.5 + 37.2i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (16.2 + 40.5i)T + (-695. + 663. i)T^{2} \) |
| 37 | \( 1 + (16.8 - 29.1i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-4.39 - 45.9i)T + (-1.65e3 + 318. i)T^{2} \) |
| 43 | \( 1 + (35.3 + 16.1i)T + (1.21e3 + 1.39e3i)T^{2} \) |
| 47 | \( 1 + (-2.13 - 8.79i)T + (-1.96e3 + 1.01e3i)T^{2} \) |
| 53 | \( 1 + (-9.61 + 4.39i)T + (1.83e3 - 2.12e3i)T^{2} \) |
| 59 | \( 1 + (4.46 - 31.0i)T + (-3.33e3 - 980. i)T^{2} \) |
| 61 | \( 1 + (17.8 + 92.7i)T + (-3.45e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (43.9 - 61.6i)T + (-1.64e3 - 4.76e3i)T^{2} \) |
| 73 | \( 1 + (-34.4 + 6.63i)T + (4.94e3 - 1.98e3i)T^{2} \) |
| 79 | \( 1 + (-68.5 - 87.1i)T + (-1.47e3 + 6.06e3i)T^{2} \) |
| 83 | \( 1 + (22.1 + 64.0i)T + (-5.41e3 + 4.25e3i)T^{2} \) |
| 89 | \( 1 + (-34.2 + 10.0i)T + (6.66e3 - 4.28e3i)T^{2} \) |
| 97 | \( 1 + (-148. - 85.4i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−11.86627998686061718260197374986, −11.18184910002030613334915522343, −10.18001004574949249600618824481, −9.453984224242597964184135380970, −7.84386522642579122680117121812, −6.95540747916177326369213097951, −5.97469309577792572833813639186, −4.81974843783307182423508837583, −2.37281572591565489325959380043, −1.76558658489222136787841564064,
1.76284084669248702186648886917, 3.38777772748894666914180437199, 4.96276374481445504801165110887, 5.79409264662754995271492102754, 7.42569370548155734742184800171, 8.605768826124748026653082513026, 8.895474228544720859259157499288, 10.59050820035110305538613422818, 11.17037612061479108652484371438, 12.31685930599996884780018426388
