| L(s) = 1 | + (−3.20 − 3.70i)2-s + (5.16 − 0.593i)3-s + (−2.27 + 15.8i)4-s + (−8.51 − 2.50i)5-s + (−18.7 − 17.2i)6-s + (14.3 − 12.4i)7-s + (32.9 − 21.1i)8-s + (26.2 − 6.13i)9-s + (18.0 + 39.5i)10-s + (34.2 + 10.0i)11-s + (−2.34 + 83.0i)12-s + (−1.35 + 2.10i)13-s + (−92.0 − 13.2i)14-s + (−45.4 − 7.85i)15-s + (−61.1 − 17.9i)16-s + (−9.48 + 1.36i)17-s + ⋯ |
| L(s) = 1 | + (−1.13 − 1.30i)2-s + (0.993 − 0.114i)3-s + (−0.284 + 1.97i)4-s + (−0.761 − 0.223i)5-s + (−1.27 − 1.17i)6-s + (0.774 − 0.671i)7-s + (1.45 − 0.934i)8-s + (0.973 − 0.227i)9-s + (0.571 + 1.25i)10-s + (0.938 + 0.275i)11-s + (−0.0564 + 1.99i)12-s + (−0.0288 + 0.0449i)13-s + (−1.75 − 0.252i)14-s + (−0.782 − 0.135i)15-s + (−0.955 − 0.280i)16-s + (−0.135 + 0.0194i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.382253 - 1.20711i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.382253 - 1.20711i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-5.16 + 0.593i)T \) |
| 67 | \( 1 + (494. + 238. i)T \) |
| good | 2 | \( 1 + (3.20 + 3.70i)T + (-1.13 + 7.91i)T^{2} \) |
| 5 | \( 1 + (8.51 + 2.50i)T + (105. + 67.5i)T^{2} \) |
| 7 | \( 1 + (-14.3 + 12.4i)T + (48.8 - 339. i)T^{2} \) |
| 11 | \( 1 + (-34.2 - 10.0i)T + (1.11e3 + 719. i)T^{2} \) |
| 13 | \( 1 + (1.35 - 2.10i)T + (-912. - 1.99e3i)T^{2} \) |
| 17 | \( 1 + (9.48 - 1.36i)T + (4.71e3 - 1.38e3i)T^{2} \) |
| 19 | \( 1 + (-79.8 + 92.0i)T + (-976. - 6.78e3i)T^{2} \) |
| 23 | \( 1 + (19.1 + 8.75i)T + (7.96e3 + 9.19e3i)T^{2} \) |
| 29 | \( 1 - 82.2iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (159. + 247. i)T + (-1.23e4 + 2.70e4i)T^{2} \) |
| 37 | \( 1 + 96.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + (37.3 + 260. i)T + (-6.61e4 + 1.94e4i)T^{2} \) |
| 43 | \( 1 + (-216. + 31.0i)T + (7.62e4 - 2.23e4i)T^{2} \) |
| 47 | \( 1 + (194. + 88.8i)T + (6.79e4 + 7.84e4i)T^{2} \) |
| 53 | \( 1 + (24.3 - 169. i)T + (-1.42e5 - 4.19e4i)T^{2} \) |
| 59 | \( 1 + (57.4 + 89.3i)T + (-8.53e4 + 1.86e5i)T^{2} \) |
| 61 | \( 1 + (-206. - 703. i)T + (-1.90e5 + 1.22e5i)T^{2} \) |
| 71 | \( 1 + (-571. - 82.2i)T + (3.43e5 + 1.00e5i)T^{2} \) |
| 73 | \( 1 + (-666. + 195. i)T + (3.27e5 - 2.10e5i)T^{2} \) |
| 79 | \( 1 + (-240. + 373. i)T + (-2.04e5 - 4.48e5i)T^{2} \) |
| 83 | \( 1 + (-351. + 1.19e3i)T + (-4.81e5 - 3.09e5i)T^{2} \) |
| 89 | \( 1 + (-411. + 188. i)T + (4.61e5 - 5.32e5i)T^{2} \) |
| 97 | \( 1 - 206. iT - 9.12e5T^{2} \) |
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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
−11.51254901510325750208816217566, −10.61082251843532056819278083931, −9.467709235796725127018166768451, −8.872061097939764244018292690680, −7.83539954409615776345606519393, −7.24480244819984227440933377697, −4.37171837608866128046781126661, −3.51886710788015933556691230636, −2.02050438811889944643478211521, −0.799910695190741436931457487938,
1.50680481132969859139686383587, 3.62737819931806861219463577064, 5.24641457956497309993458463167, 6.63562708455976294687376445378, 7.70814688167293834501420545795, 8.213922591225625491636733373544, 9.082296818862284212212928390952, 9.869782265443178280620886678316, 11.18497981350824606462653001923, 12.29407703925439987170373214166
