| L(s) = 1 | + (−0.956 − 0.0110i)2-s + (0.269 − 1.18i)3-s + (−1.08 − 0.0249i)4-s + (2.02 − 3.79i)5-s + (−0.271 + 1.12i)6-s + (0.434 + 1.07i)7-s + (2.94 + 0.101i)8-s + (1.37 + 0.663i)9-s + (−1.97 + 3.60i)10-s + (2.66 + 2.77i)11-s + (−0.322 + 1.27i)12-s + (5.72 + 0.862i)13-s + (−0.403 − 1.02i)14-s + (−3.94 − 3.41i)15-s + (−0.649 − 0.0298i)16-s + (0.162 − 0.964i)17-s + ⋯ |
| L(s) = 1 | + (−0.676 − 0.00778i)2-s + (0.155 − 0.682i)3-s + (−0.542 − 0.0124i)4-s + (0.903 − 1.69i)5-s + (−0.110 + 0.460i)6-s + (0.164 + 0.404i)7-s + (1.04 + 0.0360i)8-s + (0.459 + 0.221i)9-s + (−0.624 + 1.14i)10-s + (0.803 + 0.836i)11-s + (−0.0931 + 0.368i)12-s + (1.58 + 0.239i)13-s + (−0.107 − 0.274i)14-s + (−1.01 − 0.881i)15-s + (−0.162 − 0.00747i)16-s + (0.0394 − 0.233i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05523 - 0.748624i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05523 - 0.748624i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 547 | \( 1 + (6.91 + 22.3i)T \) |
| good | 2 | \( 1 + (0.956 + 0.0110i)T + (1.99 + 0.0460i)T^{2} \) |
| 3 | \( 1 + (-0.269 + 1.18i)T + (-2.70 - 1.30i)T^{2} \) |
| 5 | \( 1 + (-2.02 + 3.79i)T + (-2.79 - 4.14i)T^{2} \) |
| 7 | \( 1 + (-0.434 - 1.07i)T + (-5.02 + 4.87i)T^{2} \) |
| 11 | \( 1 + (-2.66 - 2.77i)T + (-0.442 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-5.72 - 0.862i)T + (12.4 + 3.83i)T^{2} \) |
| 17 | \( 1 + (-0.162 + 0.964i)T + (-16.0 - 5.56i)T^{2} \) |
| 19 | \( 1 + (1.26 - 0.684i)T + (10.4 - 15.8i)T^{2} \) |
| 23 | \( 1 + (-3.51 - 0.822i)T + (20.6 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-3.15 - 1.79i)T + (14.7 + 24.9i)T^{2} \) |
| 31 | \( 1 + (9.51 + 3.42i)T + (23.8 + 19.7i)T^{2} \) |
| 37 | \( 1 + (6.15 - 4.96i)T + (7.81 - 36.1i)T^{2} \) |
| 41 | \( 1 + (-1.50 - 2.61i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.673 - 6.86i)T + (-42.1 - 8.35i)T^{2} \) |
| 47 | \( 1 + (5.30 - 0.428i)T + (46.3 - 7.53i)T^{2} \) |
| 53 | \( 1 + (-4.59 + 1.59i)T + (41.6 - 32.8i)T^{2} \) |
| 59 | \( 1 + (9.72 - 4.14i)T + (40.8 - 42.5i)T^{2} \) |
| 61 | \( 1 + (-2.02 + 0.948i)T + (39.1 - 46.8i)T^{2} \) |
| 67 | \( 1 + (-5.43 - 5.28i)T + (1.92 + 66.9i)T^{2} \) |
| 71 | \( 1 + (10.3 - 2.05i)T + (65.6 - 27.0i)T^{2} \) |
| 73 | \( 1 + (2.80 - 2.42i)T + (10.4 - 72.2i)T^{2} \) |
| 79 | \( 1 + (7.64 + 5.47i)T + (25.4 + 74.7i)T^{2} \) |
| 83 | \( 1 + (10.7 + 6.81i)T + (35.5 + 74.9i)T^{2} \) |
| 89 | \( 1 + (-0.490 - 5.66i)T + (-87.6 + 15.2i)T^{2} \) |
| 97 | \( 1 + (1.70 + 12.7i)T + (-93.6 + 25.3i)T^{2} \) |
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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
−10.25350490120110834180476936293, −9.473308373069174718421096109047, −8.777596649516134702003103921705, −8.401194158232306975512479824961, −7.21034088382750322225057175690, −6.01834461485005750156714179298, −4.95879325557472903325281307540, −4.17742683912735450633264161634, −1.67668939462479366438277339072, −1.29482832006093192187037045604,
1.48073719835119379912803652124, 3.37871228924193863770556636863, 3.94018565571005723523261257432, 5.55527039337829344984753007977, 6.56494491755915867518894838566, 7.32766138014317562068858093857, 8.727702994319623657461927203961, 9.185895807994231960729643395993, 10.22809086829852787360653009147, 10.71097833411091263714470138246
