| 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 | \( 1 + (1.71 - 0.623i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.86189695171103708608780261804, −11.06269053986771194505433996398, −9.677262081395484123732093756283, −8.842365091917866799198863105656, −8.064949941231108301184835933670, −7.01488845312336017854780117710, −5.30437806294735576724670489606, −4.48661689939870388590103318892, −3.99442793144533782518383509248, −2.14852530416668960944018930722,
0.73518401232791650125106014266, 3.24299132548125077407655519840, 4.08766857078494770068511858602, 5.29658758404962683202688321319, 6.36595314025299148760480201408, 7.39941863348587348913837505218, 8.096532675352867446873664808197, 9.461820327075300824025851450452, 10.59248602402382248187186769543, 11.02759612231929568334888673680
