| L(s) = 1 | + (0.893 − 0.828i)2-s + (−1.81 + 0.273i)3-s + (−0.0385 + 0.513i)4-s + (0.176 − 0.450i)5-s + (−1.39 + 1.75i)6-s + (1.91 + 2.39i)8-s + (0.357 − 0.110i)9-s + (−0.215 − 0.548i)10-s + (3.34 + 1.03i)11-s + (−0.0707 − 0.943i)12-s + (0.866 + 3.79i)13-s + (−0.197 + 0.866i)15-s + (2.67 + 0.403i)16-s + (0.278 − 0.189i)17-s + (0.228 − 0.395i)18-s + (3.24 + 5.62i)19-s + ⋯ |
| L(s) = 1 | + (0.631 − 0.586i)2-s + (−1.04 + 0.158i)3-s + (−0.0192 + 0.256i)4-s + (0.0790 − 0.201i)5-s + (−0.569 + 0.714i)6-s + (0.675 + 0.847i)8-s + (0.119 − 0.0367i)9-s + (−0.0681 − 0.173i)10-s + (1.00 + 0.311i)11-s + (−0.0204 − 0.272i)12-s + (0.240 + 1.05i)13-s + (−0.0510 + 0.223i)15-s + (0.668 + 0.100i)16-s + (0.0674 − 0.0460i)17-s + (0.0537 − 0.0931i)18-s + (0.744 + 1.29i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29311 + 0.346481i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29311 + 0.346481i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + (-0.893 + 0.828i)T + (0.149 - 1.99i)T^{2} \) |
| 3 | \( 1 + (1.81 - 0.273i)T + (2.86 - 0.884i)T^{2} \) |
| 5 | \( 1 + (-0.176 + 0.450i)T + (-3.66 - 3.40i)T^{2} \) |
| 11 | \( 1 + (-3.34 - 1.03i)T + (9.08 + 6.19i)T^{2} \) |
| 13 | \( 1 + (-0.866 - 3.79i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (-0.278 + 0.189i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (-3.24 - 5.62i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.87 + 1.27i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (5.84 + 2.81i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (-2.13 + 3.69i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.417 + 5.57i)T + (-36.5 + 5.51i)T^{2} \) |
| 41 | \( 1 + (2.67 + 3.35i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (5.28 - 6.63i)T + (-9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (3.61 - 3.35i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (-0.673 + 8.98i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (-4.54 - 11.5i)T + (-43.2 + 40.1i)T^{2} \) |
| 61 | \( 1 + (0.683 + 9.12i)T + (-60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (2.35 - 4.07i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-12.8 + 6.20i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (5.92 + 5.49i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-0.516 - 0.893i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.93 + 12.8i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-1.80 + 0.555i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 + 0.104T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.67507163508324796096047028832, −11.15171939985462081097304475302, −9.988931713574836127233963179979, −8.945958352813712966556283586144, −7.74409921984248647386209646674, −6.51610874766779547626037381809, −5.51221380804028609977708493554, −4.49116911718419128933011050944, −3.58416819589413384496768452207, −1.77286336643324258550149987788,
0.974622653789998146491233866739, 3.38118018951476271126744286651, 4.87368475849252081255811444098, 5.57293330525748412060291254549, 6.47187414415511907672921122938, 7.07380530231195958497036790352, 8.560821688288528688432723570023, 9.782559700210348715446921997255, 10.71308819123778787675640936260, 11.44944175211716282871594598798
