| L(s) = 1 | + (−1.48 − 1.57i)2-s + (−1.47 − 0.900i)3-s + (−0.155 + 2.66i)4-s + (−1.95 + 0.228i)5-s + (0.777 + 3.66i)6-s + (−0.320 − 0.160i)7-s + (1.10 − 0.928i)8-s + (1.37 + 2.66i)9-s + (3.25 + 2.72i)10-s + (−0.894 − 2.07i)11-s + (2.62 − 3.80i)12-s + (−6.78 − 1.60i)13-s + (0.222 + 0.741i)14-s + (3.09 + 1.42i)15-s + (2.20 + 0.257i)16-s + (0.396 − 2.25i)17-s + ⋯ |
| L(s) = 1 | + (−1.04 − 1.11i)2-s + (−0.854 − 0.520i)3-s + (−0.0775 + 1.33i)4-s + (−0.872 + 0.101i)5-s + (0.317 + 1.49i)6-s + (−0.121 − 0.0607i)7-s + (0.391 − 0.328i)8-s + (0.459 + 0.888i)9-s + (1.02 + 0.862i)10-s + (−0.269 − 0.625i)11-s + (0.758 − 1.09i)12-s + (−1.88 − 0.445i)13-s + (0.0593 + 0.198i)14-s + (0.798 + 0.366i)15-s + (0.550 + 0.0643i)16-s + (0.0962 − 0.546i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0588442 + 0.157805i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0588442 + 0.157805i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.47 + 0.900i)T \) |
| good | 2 | \( 1 + (1.48 + 1.57i)T + (-0.116 + 1.99i)T^{2} \) |
| 5 | \( 1 + (1.95 - 0.228i)T + (4.86 - 1.15i)T^{2} \) |
| 7 | \( 1 + (0.320 + 0.160i)T + (4.18 + 5.61i)T^{2} \) |
| 11 | \( 1 + (0.894 + 2.07i)T + (-7.54 + 8.00i)T^{2} \) |
| 13 | \( 1 + (6.78 + 1.60i)T + (11.6 + 5.83i)T^{2} \) |
| 17 | \( 1 + (-0.396 + 2.25i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (1.11 + 6.34i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (-2.66 + 1.33i)T + (13.7 - 18.4i)T^{2} \) |
| 29 | \( 1 + (1.97 - 6.60i)T + (-24.2 - 15.9i)T^{2} \) |
| 31 | \( 1 + (-2.70 - 1.78i)T + (12.2 + 28.4i)T^{2} \) |
| 37 | \( 1 + (-7.54 + 2.74i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (1.48 - 1.56i)T + (-2.38 - 40.9i)T^{2} \) |
| 43 | \( 1 + (2.40 - 3.23i)T + (-12.3 - 41.1i)T^{2} \) |
| 47 | \( 1 + (-4.29 + 2.82i)T + (18.6 - 43.1i)T^{2} \) |
| 53 | \( 1 + (2.78 - 4.82i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.03 + 11.6i)T + (-40.4 - 42.9i)T^{2} \) |
| 61 | \( 1 + (0.123 + 2.12i)T + (-60.5 + 7.08i)T^{2} \) |
| 67 | \( 1 + (-1.02 - 3.42i)T + (-55.9 + 36.8i)T^{2} \) |
| 71 | \( 1 + (10.2 + 8.59i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-1.01 + 0.850i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (1.41 + 1.49i)T + (-4.59 + 78.8i)T^{2} \) |
| 83 | \( 1 + (-4.49 - 4.76i)T + (-4.82 + 82.8i)T^{2} \) |
| 89 | \( 1 + (-0.344 + 0.289i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (9.74 + 1.13i)T + (94.3 + 22.3i)T^{2} \) |
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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
−13.09331877251831659147354703965, −12.19494781945408131205445453911, −11.40264519036238458674935986534, −10.66429893716839406208927115056, −9.478806342849838769897713377023, −8.026477487075114127218855793745, −7.06939157283196114708006600554, −5.02550434941107152618809870145, −2.75638999681381883807236768823, −0.31886031586688489646151823194,
4.34300810695175964535428837500, 5.83044054406207264816006362225, 7.15085442037796890204160108076, 8.002096581861220875333815647045, 9.567667748734323325972057227438, 10.15880333034800388882077553099, 11.75643491120748608053817452725, 12.51541925197986152783820413742, 14.81013320676862956823993564667, 15.20094929236941803473249971461
