| L(s) = 1 | + (−1.57 − 1.99i)2-s + (1.54 + 0.780i)3-s + (−1.03 + 4.30i)4-s + (−0.347 + 1.90i)5-s + (−0.875 − 4.30i)6-s + (0.530 − 0.293i)7-s + (5.59 − 2.58i)8-s + (1.78 + 2.41i)9-s + (4.33 − 2.29i)10-s + (−1.87 − 1.22i)11-s + (−4.95 + 5.85i)12-s + (−1.29 + 1.18i)13-s + (−1.41 − 0.594i)14-s + (−2.02 + 2.67i)15-s + (−6.03 − 3.06i)16-s + (−1.21 − 2.02i)17-s + ⋯ |
| L(s) = 1 | + (−1.11 − 1.40i)2-s + (0.892 + 0.450i)3-s + (−0.515 + 2.15i)4-s + (−0.155 + 0.851i)5-s + (−0.357 − 1.75i)6-s + (0.200 − 0.110i)7-s + (1.97 − 0.914i)8-s + (0.593 + 0.804i)9-s + (1.37 − 0.726i)10-s + (−0.566 − 0.369i)11-s + (−1.43 + 1.69i)12-s + (−0.359 + 0.328i)13-s + (−0.378 − 0.158i)14-s + (−0.522 + 0.690i)15-s + (−1.50 − 0.765i)16-s + (−0.295 − 0.490i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 - 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.854 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.840189 + 0.235651i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.840189 + 0.235651i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.54 - 0.780i)T \) |
| 59 | \( 1 + (-7.67 - 0.314i)T \) |
| good | 2 | \( 1 + (1.57 + 1.99i)T + (-0.465 + 1.94i)T^{2} \) |
| 5 | \( 1 + (0.347 - 1.90i)T + (-4.67 - 1.76i)T^{2} \) |
| 7 | \( 1 + (-0.530 + 0.293i)T + (3.71 - 5.93i)T^{2} \) |
| 11 | \( 1 + (1.87 + 1.22i)T + (4.43 + 10.0i)T^{2} \) |
| 13 | \( 1 + (1.29 - 1.18i)T + (1.17 - 12.9i)T^{2} \) |
| 17 | \( 1 + (1.21 + 2.02i)T + (-7.96 + 15.0i)T^{2} \) |
| 19 | \( 1 + (1.95 - 7.03i)T + (-16.2 - 9.79i)T^{2} \) |
| 23 | \( 1 + (-0.834 + 1.72i)T + (-14.2 - 18.0i)T^{2} \) |
| 29 | \( 1 + (0.952 - 6.54i)T + (-27.7 - 8.26i)T^{2} \) |
| 31 | \( 1 + (-5.02 - 4.93i)T + (0.559 + 30.9i)T^{2} \) |
| 37 | \( 1 + (2.09 - 4.53i)T + (-23.9 - 28.1i)T^{2} \) |
| 41 | \( 1 + (-0.0251 + 0.347i)T + (-40.5 - 5.90i)T^{2} \) |
| 43 | \( 1 + (0.757 - 1.49i)T + (-25.4 - 34.6i)T^{2} \) |
| 47 | \( 1 + (5.28 - 0.964i)T + (43.9 - 16.6i)T^{2} \) |
| 53 | \( 1 + (2.02 + 1.07i)T + (29.7 + 43.8i)T^{2} \) |
| 61 | \( 1 + (-11.8 + 9.35i)T + (14.1 - 59.3i)T^{2} \) |
| 67 | \( 1 + (0.209 - 0.297i)T + (-22.5 - 63.0i)T^{2} \) |
| 71 | \( 1 + (-2.39 + 2.03i)T + (11.4 - 70.0i)T^{2} \) |
| 73 | \( 1 + (8.08 + 10.6i)T + (-19.5 + 70.3i)T^{2} \) |
| 79 | \( 1 + (-5.23 + 11.8i)T + (-53.2 - 58.3i)T^{2} \) |
| 83 | \( 1 + (-0.0455 + 0.226i)T + (-76.5 - 32.1i)T^{2} \) |
| 89 | \( 1 + (-0.730 - 1.83i)T + (-64.6 + 61.2i)T^{2} \) |
| 97 | \( 1 + (-0.244 - 0.583i)T + (-67.9 + 69.2i)T^{2} \) |
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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
−10.57788358882025543580691606496, −10.25378959397658570267099650829, −9.334612398842289102710923866042, −8.438995974254876349742146301347, −7.87682330699494515110983654113, −6.83075938754433664246285414737, −4.78384565419673741611362173290, −3.50395744779540186243098107418, −2.89041179917845561582376720243, −1.75441490031071684689649793930,
0.66984524885992744079177381087, 2.30643016420406108652986665375, 4.40322018353285703680809724103, 5.40132276928560079538734855831, 6.61938571773105819678408915180, 7.35982832747810373392375305223, 8.240551107505703208662953439647, 8.614223571480920620302264013771, 9.491292477569976686341408229019, 10.17047201097741925079552862184
