| L(s) = 1 | + (0.782 + 0.311i)2-s + (−1.72 − 0.187i)3-s + (−2.38 − 2.26i)4-s + (−2.21 + 2.60i)5-s + (−1.28 − 0.683i)6-s + (9.58 + 5.76i)7-s + (−2.57 − 5.57i)8-s + (2.92 + 0.644i)9-s + (−2.54 + 1.34i)10-s + (16.3 + 0.886i)11-s + (3.69 + 4.34i)12-s + (3.90 + 17.7i)13-s + (5.70 + 7.49i)14-s + (4.29 − 4.07i)15-s + (0.432 + 7.98i)16-s + (−3.56 + 2.14i)17-s + ⋯ |
| L(s) = 1 | + (0.391 + 0.155i)2-s + (−0.573 − 0.0624i)3-s + (−0.597 − 0.565i)4-s + (−0.442 + 0.521i)5-s + (−0.214 − 0.113i)6-s + (1.36 + 0.824i)7-s + (−0.322 − 0.696i)8-s + (0.325 + 0.0716i)9-s + (−0.254 + 0.134i)10-s + (1.48 + 0.0806i)11-s + (0.307 + 0.362i)12-s + (0.300 + 1.36i)13-s + (0.407 + 0.535i)14-s + (0.286 − 0.271i)15-s + (0.0270 + 0.499i)16-s + (−0.209 + 0.126i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.34002 + 0.504936i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34002 + 0.504936i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.72 + 0.187i)T \) |
| 59 | \( 1 + (46.9 + 35.7i)T \) |
| good | 2 | \( 1 + (-0.782 - 0.311i)T + (2.90 + 2.75i)T^{2} \) |
| 5 | \( 1 + (2.21 - 2.60i)T + (-4.04 - 24.6i)T^{2} \) |
| 7 | \( 1 + (-9.58 - 5.76i)T + (22.9 + 43.2i)T^{2} \) |
| 11 | \( 1 + (-16.3 - 0.886i)T + (120. + 13.0i)T^{2} \) |
| 13 | \( 1 + (-3.90 - 17.7i)T + (-153. + 70.9i)T^{2} \) |
| 17 | \( 1 + (3.56 - 2.14i)T + (135. - 255. i)T^{2} \) |
| 19 | \( 1 + (-1.67 + 6.03i)T + (-309. - 186. i)T^{2} \) |
| 23 | \( 1 + (-19.0 + 12.9i)T + (195. - 491. i)T^{2} \) |
| 29 | \( 1 + (6.81 + 17.1i)T + (-610. + 578. i)T^{2} \) |
| 31 | \( 1 + (24.4 - 6.79i)T + (823. - 495. i)T^{2} \) |
| 37 | \( 1 + (8.36 - 18.0i)T + (-886. - 1.04e3i)T^{2} \) |
| 41 | \( 1 + (-10.4 + 15.4i)T + (-622. - 1.56e3i)T^{2} \) |
| 43 | \( 1 + (-64.8 + 3.51i)T + (1.83e3 - 199. i)T^{2} \) |
| 47 | \( 1 + (52.4 - 44.5i)T + (357. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + (-31.0 + 58.5i)T + (-1.57e3 - 2.32e3i)T^{2} \) |
| 61 | \( 1 + (-38.4 - 15.3i)T + (2.70e3 + 2.55e3i)T^{2} \) |
| 67 | \( 1 + (-32.1 - 69.5i)T + (-2.90e3 + 3.42e3i)T^{2} \) |
| 71 | \( 1 + (48.9 + 57.6i)T + (-815. + 4.97e3i)T^{2} \) |
| 73 | \( 1 + (53.6 + 70.5i)T + (-1.42e3 + 5.13e3i)T^{2} \) |
| 79 | \( 1 + (-14.9 + 1.62i)T + (6.09e3 - 1.34e3i)T^{2} \) |
| 83 | \( 1 + (-9.68 - 28.7i)T + (-5.48e3 + 4.16e3i)T^{2} \) |
| 89 | \( 1 + (-64.5 + 25.7i)T + (5.75e3 - 5.44e3i)T^{2} \) |
| 97 | \( 1 + (51.1 - 67.2i)T + (-2.51e3 - 9.06e3i)T^{2} \) |
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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
−12.41260515340470710809807441082, −11.45997546350129576000633167722, −11.04057961595517186362406280044, −9.388507151044597750781653023216, −8.710908182961647319839385760260, −7.07340243203614298548028867142, −6.13663174918706928635401386514, −4.91267063616637399819936420005, −4.03734138360011387025135977361, −1.55612187074294025859808711296,
1.01497078716767894942587346157, 3.70936028617703402120290787647, 4.53817976121147452725856585580, 5.53744801530769174829071887763, 7.31735132388865031872165064330, 8.214214338074832436391371975591, 9.166579440973260349604449195343, 10.73546918543068068071563042637, 11.48899121693492788289956661786, 12.29795108596072352143334475315
