| L(s) = 1 | + (0.454 − 0.856i)2-s + (0.595 + 0.877i)4-s + (−3.60 + 0.792i)5-s + (1.56 − 1.19i)7-s + (2.94 − 0.320i)8-s + (−0.956 + 3.44i)10-s + (0.731 + 1.83i)11-s + (0.326 + 6.01i)13-s + (−0.309 − 1.88i)14-s + (0.279 − 0.701i)16-s + (1.98 + 1.50i)17-s + (−0.510 + 0.172i)19-s + (−2.83 − 2.68i)20-s + (1.90 + 0.207i)22-s + (2.30 − 1.38i)23-s + ⋯ |
| L(s) = 1 | + (0.321 − 0.605i)2-s + (0.297 + 0.438i)4-s + (−1.61 + 0.354i)5-s + (0.592 − 0.450i)7-s + (1.04 − 0.113i)8-s + (−0.302 + 1.08i)10-s + (0.220 + 0.553i)11-s + (0.0904 + 1.66i)13-s + (−0.0825 − 0.503i)14-s + (0.0698 − 0.175i)16-s + (0.480 + 0.365i)17-s + (−0.117 + 0.0394i)19-s + (−0.634 − 0.601i)20-s + (0.405 + 0.0441i)22-s + (0.481 − 0.289i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.495i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.868 - 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.49432 + 0.396576i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.49432 + 0.396576i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + (3.58 - 6.79i)T \) |
| good | 2 | \( 1 + (-0.454 + 0.856i)T + (-1.12 - 1.65i)T^{2} \) |
| 5 | \( 1 + (3.60 - 0.792i)T + (4.53 - 2.09i)T^{2} \) |
| 7 | \( 1 + (-1.56 + 1.19i)T + (1.87 - 6.74i)T^{2} \) |
| 11 | \( 1 + (-0.731 - 1.83i)T + (-7.98 + 7.56i)T^{2} \) |
| 13 | \( 1 + (-0.326 - 6.01i)T + (-12.9 + 1.40i)T^{2} \) |
| 17 | \( 1 + (-1.98 - 1.50i)T + (4.54 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.510 - 0.172i)T + (15.1 - 11.4i)T^{2} \) |
| 23 | \( 1 + (-2.30 + 1.38i)T + (10.7 - 20.3i)T^{2} \) |
| 29 | \( 1 + (-4.68 - 8.84i)T + (-16.2 + 24.0i)T^{2} \) |
| 31 | \( 1 + (6.98 + 2.35i)T + (24.6 + 18.7i)T^{2} \) |
| 37 | \( 1 + (-8.07 - 0.878i)T + (36.1 + 7.95i)T^{2} \) |
| 41 | \( 1 + (4.71 + 2.83i)T + (19.2 + 36.2i)T^{2} \) |
| 43 | \( 1 + (-1.43 + 3.59i)T + (-31.2 - 29.5i)T^{2} \) |
| 47 | \( 1 + (-1.10 - 0.242i)T + (42.6 + 19.7i)T^{2} \) |
| 53 | \( 1 + (2.40 + 8.65i)T + (-45.4 + 27.3i)T^{2} \) |
| 61 | \( 1 + (-4.54 + 8.56i)T + (-34.2 - 50.4i)T^{2} \) |
| 67 | \( 1 + (10.7 - 1.16i)T + (65.4 - 14.4i)T^{2} \) |
| 71 | \( 1 + (1.03 + 0.227i)T + (64.4 + 29.8i)T^{2} \) |
| 73 | \( 1 + (1.59 + 9.75i)T + (-69.1 + 23.3i)T^{2} \) |
| 79 | \( 1 + (6.95 + 6.58i)T + (4.27 + 78.8i)T^{2} \) |
| 83 | \( 1 + (-2.31 + 2.72i)T + (-13.4 - 81.9i)T^{2} \) |
| 89 | \( 1 + (2.07 + 3.92i)T + (-49.9 + 73.6i)T^{2} \) |
| 97 | \( 1 + (1.55 - 9.46i)T + (-91.9 - 30.9i)T^{2} \) |
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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
−11.07982849314668533008963921003, −10.54515156516720566583677035300, −9.056237603696204235152610890229, −8.060537854362900841932867764940, −7.30184064553090176234568489620, −6.76202748016641716331468892896, −4.66803779165222985769346079830, −4.11942207206821724978466722443, −3.24115862823907914315956724824, −1.68447755583523832629674194616,
0.906284150456967801068148982886, 2.99374574877666205637145053346, 4.27885627009580179039373701372, 5.20987423863618649163929039842, 6.01304848688550535461730324765, 7.36968919996262252923900509720, 7.913825272402305594929645145208, 8.587441269786292431999160385498, 9.989902553088081060013967924019, 11.07385115598829578751679535725
