| L(s) = 1 | + (−0.289 + 3.30i)2-s + (−2.26 + 1.97i)3-s + (−6.90 − 1.21i)4-s + (2.82 − 4.12i)5-s + (−5.86 − 8.04i)6-s + (−7.54 − 5.28i)7-s + (2.59 − 9.67i)8-s + (1.23 − 8.91i)9-s + (12.8 + 10.5i)10-s + (−13.3 − 4.85i)11-s + (18.0 − 10.8i)12-s + (1.78 + 20.3i)13-s + (19.6 − 23.4i)14-s + (1.72 + 14.9i)15-s + (4.85 + 1.76i)16-s + (−1.84 − 6.90i)17-s + ⋯ |
| L(s) = 1 | + (−0.144 + 1.65i)2-s + (−0.754 + 0.656i)3-s + (−1.72 − 0.304i)4-s + (0.565 − 0.824i)5-s + (−0.976 − 1.34i)6-s + (−1.07 − 0.754i)7-s + (0.323 − 1.20i)8-s + (0.137 − 0.990i)9-s + (1.28 + 1.05i)10-s + (−1.21 − 0.441i)11-s + (1.50 − 0.904i)12-s + (0.137 + 1.56i)13-s + (1.40 − 1.67i)14-s + (0.115 + 0.993i)15-s + (0.303 + 0.110i)16-s + (−0.108 − 0.406i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 + 0.863i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.505 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0335948 - 0.0192635i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0335948 - 0.0192635i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.26 - 1.97i)T \) |
| 5 | \( 1 + (-2.82 + 4.12i)T \) |
| good | 2 | \( 1 + (0.289 - 3.30i)T + (-3.93 - 0.694i)T^{2} \) |
| 7 | \( 1 + (7.54 + 5.28i)T + (16.7 + 46.0i)T^{2} \) |
| 11 | \( 1 + (13.3 + 4.85i)T + (92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (-1.78 - 20.3i)T + (-166. + 29.3i)T^{2} \) |
| 17 | \( 1 + (1.84 + 6.90i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (5.44 - 3.14i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (11.4 - 7.98i)T + (180. - 497. i)T^{2} \) |
| 29 | \( 1 + (15.3 + 18.2i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (2.53 - 14.3i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (-10.8 - 40.6i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (37.1 + 31.1i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (21.6 + 46.5i)T + (-1.18e3 + 1.41e3i)T^{2} \) |
| 47 | \( 1 + (16.4 + 11.5i)T + (755. + 2.07e3i)T^{2} \) |
| 53 | \( 1 + (-34.0 - 34.0i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-1.34 - 3.70i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-19.1 - 108. i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (101. - 8.86i)T + (4.42e3 - 779. i)T^{2} \) |
| 71 | \( 1 + (-25.4 + 44.1i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (7.97 - 29.7i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-48.7 - 58.0i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-23.3 - 2.04i)T + (6.78e3 + 1.19e3i)T^{2} \) |
| 89 | \( 1 + (-131. + 75.7i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (112. - 52.5i)T + (6.04e3 - 7.20e3i)T^{2} \) |
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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
−13.37269713996587880326934666547, −11.90636110786518277339286112815, −10.32623024321840141263107277953, −9.512000343102363833760591253001, −8.611717891480717119471175397305, −7.08841139634635305886996267011, −6.19665447957357775310327764414, −5.27492842980690485452027636509, −4.16236172473447698341201514178, −0.02800382668717324678829049818,
2.19059918273341582287796457762, 3.12177371889756389233556030423, 5.32716473350072686296086074361, 6.41287355678235574203058281428, 7.979470976954508678563471844404, 9.608917260399455714378619694913, 10.41373828841919279216180933707, 10.95262391336640796485777653912, 12.21374151700579453972338417281, 13.08808345152808733980027490182
