| L(s) = 1 | + (0.826 + 0.300i)2-s + (−0.0923 − 0.524i)3-s + (−0.939 − 0.788i)4-s + (−1.93 + 1.62i)5-s + (0.0812 − 0.460i)6-s + (0.939 − 1.62i)7-s + (−1.41 − 2.45i)8-s + (2.55 − 0.929i)9-s + (−2.09 + 0.761i)10-s + (−1.70 − 2.95i)11-s + (−0.326 + 0.565i)12-s + (0.918 − 5.21i)13-s + (1.26 − 1.06i)14-s + (1.03 + 0.866i)15-s + (−0.00727 − 0.0412i)16-s + (−1.55 − 0.565i)17-s + ⋯ |
| L(s) = 1 | + (0.584 + 0.212i)2-s + (−0.0533 − 0.302i)3-s + (−0.469 − 0.394i)4-s + (−0.867 + 0.727i)5-s + (0.0331 − 0.188i)6-s + (0.355 − 0.615i)7-s + (−0.501 − 0.868i)8-s + (0.851 − 0.309i)9-s + (−0.661 + 0.240i)10-s + (−0.514 − 0.890i)11-s + (−0.0942 + 0.163i)12-s + (0.254 − 1.44i)13-s + (0.338 − 0.283i)14-s + (0.266 + 0.223i)15-s + (−0.00181 − 0.0103i)16-s + (−0.376 − 0.137i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0540 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0540 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.880739 - 0.834313i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.880739 - 0.834313i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 + (-0.826 - 0.300i)T + (1.53 + 1.28i)T^{2} \) |
| 3 | \( 1 + (0.0923 + 0.524i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (1.93 - 1.62i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.939 + 1.62i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.70 + 2.95i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.918 + 5.21i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (1.55 + 0.565i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-1.34 - 1.13i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (3.25 - 1.18i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.971 + 1.68i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.837T + 37T^{2} \) |
| 41 | \( 1 + (-0.779 - 4.42i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-3.67 + 3.08i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (0.673 - 0.245i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-4.67 - 3.92i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (10.1 + 3.67i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-3.36 - 2.82i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-13.3 + 4.86i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-10.5 + 8.84i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (1.30 + 7.40i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-1.20 - 6.85i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (1.25 - 2.17i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.396 + 2.24i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (1.71 + 0.623i)T + (74.3 + 62.3i)T^{2} \) |
| 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
−11.02759612231929568334888673680, −10.59248602402382248187186769543, −9.461820327075300824025851450452, −8.096532675352867446873664808197, −7.39941863348587348913837505218, −6.36595314025299148760480201408, −5.29658758404962683202688321319, −4.08766857078494770068511858602, −3.24299132548125077407655519840, −0.73518401232791650125106014266,
2.14852530416668960944018930722, 3.99442793144533782518383509248, 4.48661689939870388590103318892, 5.30437806294735576724670489606, 7.01488845312336017854780117710, 8.064949941231108301184835933670, 8.842365091917866799198863105656, 9.677262081395484123732093756283, 11.06269053986771194505433996398, 11.86189695171103708608780261804
