| L(s) = 1 | + (−0.222 + 0.974i)2-s + (−1.36 − 0.421i)3-s + (−0.900 − 0.433i)4-s + (0.365 + 0.930i)5-s + (0.715 − 1.23i)6-s + (−0.610 − 1.05i)7-s + (0.623 − 0.781i)8-s + (−0.786 − 0.536i)9-s + (−0.988 + 0.149i)10-s + (1.89 − 0.911i)11-s + (1.04 + 0.973i)12-s + (4.15 + 0.626i)13-s + (1.16 − 0.359i)14-s + (−0.106 − 1.42i)15-s + (0.623 + 0.781i)16-s + (−1.81 + 4.63i)17-s + ⋯ |
| L(s) = 1 | + (−0.157 + 0.689i)2-s + (−0.789 − 0.243i)3-s + (−0.450 − 0.216i)4-s + (0.163 + 0.416i)5-s + (0.292 − 0.505i)6-s + (−0.230 − 0.399i)7-s + (0.220 − 0.276i)8-s + (−0.262 − 0.178i)9-s + (−0.312 + 0.0471i)10-s + (0.570 − 0.274i)11-s + (0.302 + 0.280i)12-s + (1.15 + 0.173i)13-s + (0.311 − 0.0961i)14-s + (−0.0276 − 0.368i)15-s + (0.155 + 0.195i)16-s + (−0.441 + 1.12i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.955574 + 0.168956i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.955574 + 0.168956i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.222 - 0.974i)T \) |
| 5 | \( 1 + (-0.365 - 0.930i)T \) |
| 43 | \( 1 + (-6.37 - 1.55i)T \) |
| good | 3 | \( 1 + (1.36 + 0.421i)T + (2.47 + 1.68i)T^{2} \) |
| 7 | \( 1 + (0.610 + 1.05i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.89 + 0.911i)T + (6.85 - 8.60i)T^{2} \) |
| 13 | \( 1 + (-4.15 - 0.626i)T + (12.4 + 3.83i)T^{2} \) |
| 17 | \( 1 + (1.81 - 4.63i)T + (-12.4 - 11.5i)T^{2} \) |
| 19 | \( 1 + (-6.84 + 4.66i)T + (6.94 - 17.6i)T^{2} \) |
| 23 | \( 1 + (-0.331 + 4.42i)T + (-22.7 - 3.42i)T^{2} \) |
| 29 | \( 1 + (-3.67 + 1.13i)T + (23.9 - 16.3i)T^{2} \) |
| 31 | \( 1 + (-4.02 - 3.73i)T + (2.31 + 30.9i)T^{2} \) |
| 37 | \( 1 + (0.0837 - 0.145i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.370 + 1.62i)T + (-36.9 - 17.7i)T^{2} \) |
| 47 | \( 1 + (-9.29 - 4.47i)T + (29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (11.0 - 1.66i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (5.67 + 7.11i)T + (-13.1 + 57.5i)T^{2} \) |
| 61 | \( 1 + (-3.44 + 3.19i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (1.42 - 0.970i)T + (24.4 - 62.3i)T^{2} \) |
| 71 | \( 1 + (0.140 + 1.86i)T + (-70.2 + 10.5i)T^{2} \) |
| 73 | \( 1 + (10.0 + 1.51i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (6.54 + 11.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (14.7 + 4.53i)T + (68.5 + 46.7i)T^{2} \) |
| 89 | \( 1 + (-13.9 - 4.31i)T + (73.5 + 50.1i)T^{2} \) |
| 97 | \( 1 + (0.254 - 0.122i)T + (60.4 - 75.8i)T^{2} \) |
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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
−11.09707887111489248906904922854, −10.47617163997728206361006712030, −9.226960895908532351377256626265, −8.504289128666460705063980484464, −7.21898326388478470007232207754, −6.38073072943132474620730834398, −5.94891740522175873680747970092, −4.58096186234013593291402045593, −3.26461334757106778745660643666, −0.971666121043048203462176180214,
1.14893483545490445370982516035, 2.90006236975124766727294302891, 4.22370316346716629173221587812, 5.37960814561656977024853122427, 6.05143043655521287562056830843, 7.52078543530291171040262682368, 8.678680548414830073546475466357, 9.451981723429944327420337291452, 10.25098815017646500731505196703, 11.33970585850358773883271467243
