| L(s) = 1 | + (1.56 + 1.98i)2-s + (−1.73 + 0.0219i)3-s + (−1.02 + 4.27i)4-s + (0.725 − 3.97i)5-s + (−2.75 − 3.40i)6-s + (3.27 − 1.81i)7-s + (−5.49 + 2.54i)8-s + (2.99 − 0.0761i)9-s + (9.02 − 4.78i)10-s + (−2.36 − 1.53i)11-s + (1.67 − 7.42i)12-s + (2.76 − 2.52i)13-s + (8.71 + 3.65i)14-s + (−1.16 + 6.89i)15-s + (−5.82 − 2.95i)16-s + (0.0359 + 0.0597i)17-s + ⋯ |
| L(s) = 1 | + (1.10 + 1.40i)2-s + (−0.999 + 0.0126i)3-s + (−0.511 + 2.13i)4-s + (0.324 − 1.77i)5-s + (−1.12 − 1.38i)6-s + (1.23 − 0.684i)7-s + (−1.94 + 0.899i)8-s + (0.999 − 0.0253i)9-s + (2.85 − 1.51i)10-s + (−0.711 − 0.464i)11-s + (0.483 − 2.14i)12-s + (0.766 − 0.700i)13-s + (2.33 + 0.977i)14-s + (−0.301 + 1.78i)15-s + (−1.45 − 0.738i)16-s + (0.00871 + 0.0144i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.98117 + 0.719821i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98117 + 0.719821i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.73 - 0.0219i)T \) |
| 59 | \( 1 + (-5.06 - 5.77i)T \) |
| good | 2 | \( 1 + (-1.56 - 1.98i)T + (-0.465 + 1.94i)T^{2} \) |
| 5 | \( 1 + (-0.725 + 3.97i)T + (-4.67 - 1.76i)T^{2} \) |
| 7 | \( 1 + (-3.27 + 1.81i)T + (3.71 - 5.93i)T^{2} \) |
| 11 | \( 1 + (2.36 + 1.53i)T + (4.43 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.76 + 2.52i)T + (1.17 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.0359 - 0.0597i)T + (-7.96 + 15.0i)T^{2} \) |
| 19 | \( 1 + (-0.544 + 1.96i)T + (-16.2 - 9.79i)T^{2} \) |
| 23 | \( 1 + (3.18 - 6.56i)T + (-14.2 - 18.0i)T^{2} \) |
| 29 | \( 1 + (-0.347 + 2.38i)T + (-27.7 - 8.26i)T^{2} \) |
| 31 | \( 1 + (-5.95 - 5.84i)T + (0.559 + 30.9i)T^{2} \) |
| 37 | \( 1 + (0.268 - 0.579i)T + (-23.9 - 28.1i)T^{2} \) |
| 41 | \( 1 + (-0.133 + 1.84i)T + (-40.5 - 5.90i)T^{2} \) |
| 43 | \( 1 + (-3.15 + 6.22i)T + (-25.4 - 34.6i)T^{2} \) |
| 47 | \( 1 + (3.96 - 0.723i)T + (43.9 - 16.6i)T^{2} \) |
| 53 | \( 1 + (-7.18 - 3.81i)T + (29.7 + 43.8i)T^{2} \) |
| 61 | \( 1 + (3.76 - 2.97i)T + (14.1 - 59.3i)T^{2} \) |
| 67 | \( 1 + (-6.21 + 8.82i)T + (-22.5 - 63.0i)T^{2} \) |
| 71 | \( 1 + (10.8 - 9.23i)T + (11.4 - 70.0i)T^{2} \) |
| 73 | \( 1 + (-7.71 - 10.1i)T + (-19.5 + 70.3i)T^{2} \) |
| 79 | \( 1 + (2.28 - 5.17i)T + (-53.2 - 58.3i)T^{2} \) |
| 83 | \( 1 + (1.18 - 5.86i)T + (-76.5 - 32.1i)T^{2} \) |
| 89 | \( 1 + (3.19 + 8.02i)T + (-64.6 + 61.2i)T^{2} \) |
| 97 | \( 1 + (6.84 + 16.3i)T + (-67.9 + 69.2i)T^{2} \) |
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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
−11.26627106529812461298768318238, −10.09873675956914334201537058273, −8.609334326670667320749458255063, −8.068341493532000405821348294487, −7.22705185514163663107925323609, −5.81495921218575264915265936974, −5.43208319362070706622867208008, −4.72680756216183884432627717329, −3.97727663064280965224035897808, −1.13080180890172903028286626652,
1.80527565871016742054427449316, 2.58671597174255091701820921682, 4.00347786399021049457261461343, 4.92605197767880178052755414053, 5.91148813117990829066076166409, 6.55990876186710804364442433479, 7.936864738429640987696033613749, 9.704270126120396405589513464485, 10.45640613176879167928167124556, 10.92248656619886353403202492235
