| L(s) = 1 | + (−0.371 + 0.429i)2-s + (−0.415 − 0.909i)3-s + (0.238 + 1.66i)4-s + (−2.09 + 0.615i)5-s + (0.544 + 0.160i)6-s + (−1.13 + 1.30i)7-s + (−1.75 − 1.12i)8-s + (−0.654 + 0.755i)9-s + (0.515 − 1.12i)10-s + (−2.41 + 0.708i)11-s + (1.41 − 0.906i)12-s + (−1.05 + 0.680i)13-s + (−0.139 − 0.972i)14-s + (1.43 + 1.65i)15-s + (−2.08 + 0.610i)16-s + (−0.337 + 2.34i)17-s + ⋯ |
| L(s) = 1 | + (−0.263 + 0.303i)2-s + (−0.239 − 0.525i)3-s + (0.119 + 0.830i)4-s + (−0.937 + 0.275i)5-s + (0.222 + 0.0653i)6-s + (−0.428 + 0.493i)7-s + (−0.621 − 0.399i)8-s + (−0.218 + 0.251i)9-s + (0.162 − 0.356i)10-s + (−0.727 + 0.213i)11-s + (0.407 − 0.261i)12-s + (−0.293 + 0.188i)13-s + (−0.0373 − 0.259i)14-s + (0.369 + 0.426i)15-s + (−0.520 + 0.152i)16-s + (−0.0818 + 0.569i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.808 - 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.150827 + 0.463646i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.150827 + 0.463646i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.415 + 0.909i)T \) |
| 67 | \( 1 + (-7.89 - 2.17i)T \) |
| good | 2 | \( 1 + (0.371 - 0.429i)T + (-0.284 - 1.97i)T^{2} \) |
| 5 | \( 1 + (2.09 - 0.615i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (1.13 - 1.30i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (2.41 - 0.708i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (1.05 - 0.680i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (0.337 - 2.34i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-3.76 - 4.34i)T + (-2.70 + 18.8i)T^{2} \) |
| 23 | \( 1 + (0.890 + 1.94i)T + (-15.0 + 17.3i)T^{2} \) |
| 29 | \( 1 - 1.81T + 29T^{2} \) |
| 31 | \( 1 + (0.514 + 0.330i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 - 7.08T + 37T^{2} \) |
| 41 | \( 1 + (0.379 - 2.63i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-1.21 + 8.46i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + (2.70 + 5.92i)T + (-30.7 + 35.5i)T^{2} \) |
| 53 | \( 1 + (-1.47 - 10.2i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-8.08 - 5.19i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (4.48 + 1.31i)T + (51.3 + 32.9i)T^{2} \) |
| 71 | \( 1 + (-0.278 - 1.93i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (1.12 + 0.330i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (2.80 - 1.79i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (14.1 - 4.16i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (5.60 - 12.2i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + 17.5T + 97T^{2} \) |
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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
−12.49036302585274877809263951822, −12.11740779081573013921756638849, −11.14400002578422583988760340606, −9.818592781528088795138597631496, −8.479971003740058698670720744780, −7.75514679859199885933057079095, −6.98211939884375929117986726384, −5.76616912363900261029366327981, −3.99396263264838922619671191124, −2.70113534403432925970992281230,
0.45701250563463870497560182613, 2.98020227360624132090921081470, 4.52727074419177830629004776685, 5.53350397490270262586001126532, 6.93547761602621931770520423000, 8.120819817615895953100975525777, 9.427602382407292853138099615851, 10.06337697890041152077237063852, 11.17587490879678848678818067729, 11.62440872794389144123663793580
