| L(s) = 1 | + (0.396 − 1.96i)2-s + (−1.09 + 1.33i)3-s + (−3.68 − 1.55i)4-s + (1.90 + 6.27i)5-s + (2.18 + 2.68i)6-s + (1.94 − 9.78i)7-s + (−4.50 + 6.61i)8-s + (−0.585 − 2.94i)9-s + (13.0 − 1.24i)10-s + (−16.5 − 1.62i)11-s + (6.12 − 3.22i)12-s + (17.9 + 5.43i)13-s + (−18.4 − 7.69i)14-s + (−10.4 − 4.34i)15-s + (11.1 + 11.4i)16-s + (9.63 + 23.2i)17-s + ⋯ |
| L(s) = 1 | + (0.198 − 0.980i)2-s + (−0.366 + 0.446i)3-s + (−0.921 − 0.388i)4-s + (0.380 + 1.25i)5-s + (0.364 + 0.447i)6-s + (0.278 − 1.39i)7-s + (−0.563 + 0.826i)8-s + (−0.0650 − 0.326i)9-s + (1.30 − 0.124i)10-s + (−1.50 − 0.147i)11-s + (0.510 − 0.269i)12-s + (1.37 + 0.417i)13-s + (−1.31 − 0.549i)14-s + (−0.699 − 0.289i)15-s + (0.698 + 0.715i)16-s + (0.566 + 1.36i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.57304 - 0.210857i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.57304 - 0.210857i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.396 + 1.96i)T \) |
| 3 | \( 1 + (1.09 - 1.33i)T \) |
| good | 5 | \( 1 + (-1.90 - 6.27i)T + (-20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (-1.94 + 9.78i)T + (-45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (16.5 + 1.62i)T + (118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (-17.9 - 5.43i)T + (140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (-9.63 - 23.2i)T + (-204. + 204. i)T^{2} \) |
| 19 | \( 1 + (-20.9 + 11.2i)T + (200. - 300. i)T^{2} \) |
| 23 | \( 1 + (-27.9 - 18.6i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (-22.0 + 2.16i)T + (824. - 164. i)T^{2} \) |
| 31 | \( 1 + (10.2 - 10.2i)T - 961iT^{2} \) |
| 37 | \( 1 + (-50.9 - 27.2i)T + (760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (48.2 + 32.2i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (2.69 + 3.27i)T + (-360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (-9.55 - 23.0i)T + (-1.56e3 + 1.56e3i)T^{2} \) |
| 53 | \( 1 + (15.2 + 1.50i)T + (2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (-7.08 - 23.3i)T + (-2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (-52.1 + 63.5i)T + (-725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (-84.7 - 69.5i)T + (875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (64.9 + 12.9i)T + (4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (117. - 23.4i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-12.5 + 30.2i)T + (-4.41e3 - 4.41e3i)T^{2} \) |
| 83 | \( 1 + (25.7 + 48.1i)T + (-3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (-45.8 - 68.5i)T + (-3.03e3 + 7.31e3i)T^{2} \) |
| 97 | \( 1 + (-48.2 - 48.2i)T + 9.40e3iT^{2} \) |
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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
−10.90834359368522669366118429970, −10.46632902516732866272225657633, −9.852093290000110816403890166066, −8.443222955629194349091192924668, −7.29302551055950063643304311768, −6.09428609745331232015806495079, −5.03406966145356623283280893450, −3.76134566855381936581059191298, −2.99874810232230130397873951830, −1.18557230079638012071250660556,
0.888484861516534614531565280639, 2.87814616709489637157470206164, 4.98382979711479008273734875560, 5.30477690714325800446402630554, 6.06636904417345798227919368725, 7.52212635585079748059637623516, 8.354474326559711106355593526119, 8.940240626034885055635386172386, 9.953901367773514699704867573641, 11.46370799266652690877457375291
