| L(s) = 1 | + (1.83 + 0.792i)2-s + (−2.22 − 2.01i)3-s + (2.74 + 2.91i)4-s + (1.17 + 6.65i)5-s + (−2.47 − 5.46i)6-s + (−3.53 − 2.96i)7-s + (2.72 + 7.52i)8-s + (0.867 + 8.95i)9-s + (−3.12 + 13.1i)10-s + (−0.262 + 1.48i)11-s + (−0.220 − 11.9i)12-s + (−7.29 + 20.0i)13-s + (−4.14 − 8.25i)14-s + (10.8 − 17.1i)15-s + (−0.955 + 15.9i)16-s + (24.0 − 13.8i)17-s + ⋯ |
| L(s) = 1 | + (0.918 + 0.396i)2-s + (−0.740 − 0.672i)3-s + (0.685 + 0.727i)4-s + (0.234 + 1.33i)5-s + (−0.413 − 0.910i)6-s + (−0.505 − 0.423i)7-s + (0.340 + 0.940i)8-s + (0.0964 + 0.995i)9-s + (−0.312 + 1.31i)10-s + (−0.0238 + 0.135i)11-s + (−0.0183 − 0.999i)12-s + (−0.561 + 1.54i)13-s + (−0.295 − 0.589i)14-s + (0.720 − 1.14i)15-s + (−0.0597 + 0.998i)16-s + (1.41 − 0.816i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00181 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.00181 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.38402 + 1.38152i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38402 + 1.38152i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.83 - 0.792i)T \) |
| 3 | \( 1 + (2.22 + 2.01i)T \) |
| good | 5 | \( 1 + (-1.17 - 6.65i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (3.53 + 2.96i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (0.262 - 1.48i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (7.29 - 20.0i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-24.0 + 13.8i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-7.52 - 4.34i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-0.193 - 0.230i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (34.3 - 12.5i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (15.7 - 13.1i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (-51.2 + 29.5i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-9.84 + 27.0i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-11.8 - 2.09i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-24.8 + 29.6i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 13.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + (16.1 + 91.6i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-48.6 + 57.9i)T + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (32.0 - 87.9i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (5.14 - 2.96i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (13.7 - 23.7i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-99.9 + 36.3i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-80.7 + 29.4i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (15.4 + 8.90i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (3.49 - 19.8i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−12.33125200924395354790592237347, −11.55894808288803827680271579878, −10.77607105008394729932524178292, −9.625518800036853698029275875763, −7.50894660089510000593511802652, −7.12394311676746795158051946325, −6.27786273339426594660581688284, −5.20640626410448200611004704005, −3.63325947784717097710983008373, −2.23120140957191242925124691309,
0.919649687269079144757610793640, 3.15745903103317646636609168463, 4.47432591832814864784642364183, 5.59679839464807576739281682840, 5.84691182435109006577752397864, 7.77124836983373710367883640415, 9.373767967194995370924453599973, 9.964416093682638033409286198428, 11.01121614239791992748749354952, 12.19147292899209730512218191815
