L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.258 − 0.965i)3-s + (0.866 + 0.499i)4-s + (2.19 + 0.419i)5-s + (−0.499 + 0.866i)6-s + (−2.57 − 2.57i)7-s + (−0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (−2.01 − 0.973i)10-s − 5.12·11-s + (0.707 − 0.707i)12-s + (−4.44 + 1.19i)13-s + (1.81 + 3.14i)14-s + (0.973 − 2.01i)15-s + (0.500 + 0.866i)16-s + (0.673 − 2.51i)17-s + ⋯ |
L(s) = 1 | + (−0.683 − 0.183i)2-s + (0.149 − 0.557i)3-s + (0.433 + 0.249i)4-s + (0.982 + 0.187i)5-s + (−0.204 + 0.353i)6-s + (−0.971 − 0.971i)7-s + (−0.249 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.636 − 0.307i)10-s − 1.54·11-s + (0.204 − 0.204i)12-s + (−1.23 + 0.330i)13-s + (0.485 + 0.841i)14-s + (0.251 − 0.519i)15-s + (0.125 + 0.216i)16-s + (0.163 − 0.609i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0471186 - 0.509996i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0471186 - 0.509996i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 + 0.258i)T \) |
| 3 | \( 1 + (-0.258 + 0.965i)T \) |
| 5 | \( 1 + (-2.19 - 0.419i)T \) |
| 19 | \( 1 + (3.94 - 1.86i)T \) |
good | 7 | \( 1 + (2.57 + 2.57i)T + 7iT^{2} \) |
| 11 | \( 1 + 5.12T + 11T^{2} \) |
| 13 | \( 1 + (4.44 - 1.19i)T + (11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (-0.673 + 2.51i)T + (-14.7 - 8.5i)T^{2} \) |
| 23 | \( 1 + (1.45 + 5.42i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-4.79 + 8.31i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 6.03iT - 31T^{2} \) |
| 37 | \( 1 + (-4.87 + 4.87i)T - 37iT^{2} \) |
| 41 | \( 1 + (5.16 - 2.98i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.264 + 0.0709i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (6.50 - 1.74i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-9.48 + 2.54i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (3.25 + 5.63i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.80 - 10.0i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.78 + 6.65i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-5.61 + 3.24i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (6.48 + 1.73i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.09 - 7.08i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.64 - 1.64i)T - 83iT^{2} \) |
| 89 | \( 1 + (-7.97 + 13.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.01 + 0.270i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.06569179378628852256632820397, −9.828001255877850580155212366744, −8.550469290518957253836142340905, −7.60022535631912367159263863615, −6.84746022540772347906885750229, −6.08031645318397090626716596988, −4.69789771651807154244273510288, −2.94579313194509325568570429407, −2.21438224106895458845464694212, −0.31560885308172857147822943722,
2.25741822995636417764526723086, 2.98560801841593335396224357596, 4.99158241552309927710742138660, 5.63590729753016469640087325101, 6.56313144079548793556576221213, 7.80940998699540868020062291230, 8.722412125440093036330887588848, 9.485525192650334571215788190832, 10.12384934253387803665024557333, 10.62077218166006468220119582362