| L(s) = 1 | + (0.654 + 0.755i)2-s + (−1.71 − 1.09i)3-s + (−0.142 + 0.989i)4-s + (0.475 − 1.04i)5-s + (−0.289 − 2.01i)6-s + (−0.959 − 0.281i)7-s + (−0.841 + 0.540i)8-s + (0.473 + 1.03i)9-s + (1.09 − 0.322i)10-s + (3.36 − 3.88i)11-s + (1.33 − 1.53i)12-s + (−0.870 + 0.255i)13-s + (−0.415 − 0.909i)14-s + (−1.95 + 1.25i)15-s + (−0.959 − 0.281i)16-s + (−0.937 − 6.51i)17-s + ⋯ |
| L(s) = 1 | + (0.463 + 0.534i)2-s + (−0.988 − 0.635i)3-s + (−0.0711 + 0.494i)4-s + (0.212 − 0.465i)5-s + (−0.118 − 0.822i)6-s + (−0.362 − 0.106i)7-s + (−0.297 + 0.191i)8-s + (0.157 + 0.345i)9-s + (0.347 − 0.102i)10-s + (1.01 − 1.17i)11-s + (0.384 − 0.443i)12-s + (−0.241 + 0.0708i)13-s + (−0.111 − 0.243i)14-s + (−0.506 + 0.325i)15-s + (−0.239 − 0.0704i)16-s + (−0.227 − 1.58i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.417 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.906438 - 0.580941i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.906438 - 0.580941i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.654 - 0.755i)T \) |
| 7 | \( 1 + (0.959 + 0.281i)T \) |
| 23 | \( 1 + (3.59 + 3.17i)T \) |
| good | 3 | \( 1 + (1.71 + 1.09i)T + (1.24 + 2.72i)T^{2} \) |
| 5 | \( 1 + (-0.475 + 1.04i)T + (-3.27 - 3.77i)T^{2} \) |
| 11 | \( 1 + (-3.36 + 3.88i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.870 - 0.255i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.937 + 6.51i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.887 + 6.17i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-1.28 - 8.96i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-2.48 + 1.59i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-3.54 - 7.77i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-1.59 + 3.49i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (10.5 + 6.76i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 2.75T + 47T^{2} \) |
| 53 | \( 1 + (-7.77 - 2.28i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (3.19 - 0.938i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (9.21 - 5.92i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (0.772 + 0.891i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-2.58 - 2.97i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (0.646 - 4.49i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (-8.52 + 2.50i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-2.03 - 4.46i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-9.53 - 6.12i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-5.84 + 12.7i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−11.76596417050418039803393201539, −10.88936518045587972250697624916, −9.326152072247232889219886450386, −8.664830850480633089624084937668, −7.05942240719316282950190805626, −6.64553886570281312306556206741, −5.59161193447232646735115494651, −4.72802632547955399388243276995, −3.11188151528163432974429144014, −0.77514457303045944989452570585,
1.97824914807477852512212677190, 3.78332337030449266324363404005, 4.57382103059994117159302771962, 5.95473482806524103025358486984, 6.38713236556017744905073584302, 7.971286236799567871794815284973, 9.611737225450093400433108781405, 10.08952840987973477755561504827, 10.82246596480550505840044777926, 11.94340388092893761791079412637
