| L(s) = 1 | + (−0.0454 + 0.780i)2-s + (1.37 + 0.161i)4-s + (1.44 − 0.341i)5-s + (−2.26 − 3.04i)7-s + (−0.459 + 2.60i)8-s + (0.200 + 1.13i)10-s + (3.46 − 3.67i)11-s + (−4.01 − 2.01i)13-s + (2.48 − 1.63i)14-s + (0.689 + 0.163i)16-s + (4.30 + 1.56i)17-s + (4.19 − 1.52i)19-s + (2.04 − 0.238i)20-s + (2.71 + 2.87i)22-s + (2.06 − 2.77i)23-s + ⋯ |
| L(s) = 1 | + (−0.0321 + 0.551i)2-s + (0.689 + 0.0806i)4-s + (0.644 − 0.152i)5-s + (−0.857 − 1.15i)7-s + (−0.162 + 0.922i)8-s + (0.0635 + 0.360i)10-s + (1.04 − 1.10i)11-s + (−1.11 − 0.559i)13-s + (0.663 − 0.436i)14-s + (0.172 + 0.0408i)16-s + (1.04 + 0.380i)17-s + (0.962 − 0.350i)19-s + (0.456 − 0.0533i)20-s + (0.577 + 0.612i)22-s + (0.430 − 0.577i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.90185 - 0.0797339i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.90185 - 0.0797339i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (0.0454 - 0.780i)T + (-1.98 - 0.232i)T^{2} \) |
| 5 | \( 1 + (-1.44 + 0.341i)T + (4.46 - 2.24i)T^{2} \) |
| 7 | \( 1 + (2.26 + 3.04i)T + (-2.00 + 6.70i)T^{2} \) |
| 11 | \( 1 + (-3.46 + 3.67i)T + (-0.639 - 10.9i)T^{2} \) |
| 13 | \( 1 + (4.01 + 2.01i)T + (7.76 + 10.4i)T^{2} \) |
| 17 | \( 1 + (-4.30 - 1.56i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (-4.19 + 1.52i)T + (14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (-2.06 + 2.77i)T + (-6.59 - 22.0i)T^{2} \) |
| 29 | \( 1 + (-0.545 - 0.358i)T + (11.4 + 26.6i)T^{2} \) |
| 31 | \( 1 + (-0.260 - 0.604i)T + (-21.2 + 22.5i)T^{2} \) |
| 37 | \( 1 + (-0.766 + 0.642i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (0.0397 + 0.681i)T + (-40.7 + 4.75i)T^{2} \) |
| 43 | \( 1 + (2.40 + 8.04i)T + (-35.9 + 23.6i)T^{2} \) |
| 47 | \( 1 + (3.14 - 7.28i)T + (-32.2 - 34.1i)T^{2} \) |
| 53 | \( 1 + (-2.07 - 3.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.61 - 3.83i)T + (-3.43 + 58.9i)T^{2} \) |
| 61 | \( 1 + (-6.74 + 0.788i)T + (59.3 - 14.0i)T^{2} \) |
| 67 | \( 1 + (4.42 - 2.91i)T + (26.5 - 61.5i)T^{2} \) |
| 71 | \( 1 + (-1.06 - 6.03i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (0.764 - 4.33i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (0.653 - 11.2i)T + (-78.4 - 9.17i)T^{2} \) |
| 83 | \( 1 + (-0.0942 + 1.61i)T + (-82.4 - 9.63i)T^{2} \) |
| 89 | \( 1 + (-0.181 + 1.02i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-4.85 - 1.15i)T + (86.6 + 43.5i)T^{2} \) |
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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
−10.22763082158462261256919241316, −9.663743518835964899370390389885, −8.572441913460663804807743010299, −7.50630347671327050598875772723, −6.92778510749408572740941373116, −6.04461179635267916543750769850, −5.31517053066786204746982637520, −3.72323449783855723842060318087, −2.81674501571237191504494975926, −1.07675135067736490716367277639,
1.65459568489535223121141468286, 2.55786920029798900592606630099, 3.52548855518047560672420609370, 5.08638006062215733029830865378, 6.07638345726645089932792578663, 6.79921858006004515414240456319, 7.60554790908868121035791036062, 9.221383935402780556451821712042, 9.798198554030184469774684272573, 9.962273839254942969946559721704
