| L(s) = 1 | + (−0.440 + 1.34i)2-s + (0.652 + 1.60i)3-s + (−1.61 − 1.18i)4-s + (−0.484 − 0.318i)5-s + (−2.44 + 0.171i)6-s + (−3.43 + 3.23i)7-s + (2.29 − 1.64i)8-s + (−2.14 + 2.09i)9-s + (0.641 − 0.511i)10-s + (0.127 − 0.254i)11-s + (0.845 − 3.35i)12-s + (−0.136 − 0.0589i)13-s + (−2.84 − 6.03i)14-s + (0.195 − 0.986i)15-s + (1.20 + 3.81i)16-s + (−2.86 + 3.41i)17-s + ⋯ |
| L(s) = 1 | + (−0.311 + 0.950i)2-s + (0.376 + 0.926i)3-s + (−0.806 − 0.591i)4-s + (−0.216 − 0.142i)5-s + (−0.997 + 0.0698i)6-s + (−1.29 + 1.22i)7-s + (0.812 − 0.582i)8-s + (−0.716 + 0.698i)9-s + (0.203 − 0.161i)10-s + (0.0385 − 0.0766i)11-s + (0.244 − 0.969i)12-s + (−0.0378 − 0.0163i)13-s + (−0.759 − 1.61i)14-s + (0.0504 − 0.254i)15-s + (0.300 + 0.953i)16-s + (−0.695 + 0.829i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.249444 - 0.304765i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.249444 - 0.304765i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.440 - 1.34i)T \) |
| 3 | \( 1 + (-0.652 - 1.60i)T \) |
| good | 5 | \( 1 + (0.484 + 0.318i)T + (1.98 + 4.59i)T^{2} \) |
| 7 | \( 1 + (3.43 - 3.23i)T + (0.407 - 6.98i)T^{2} \) |
| 11 | \( 1 + (-0.127 + 0.254i)T + (-6.56 - 8.82i)T^{2} \) |
| 13 | \( 1 + (0.136 + 0.0589i)T + (8.92 + 9.45i)T^{2} \) |
| 17 | \( 1 + (2.86 - 3.41i)T + (-2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-3.61 + 3.03i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (-2.17 + 2.31i)T + (-1.33 - 22.9i)T^{2} \) |
| 29 | \( 1 + (3.01 + 0.352i)T + (28.2 + 6.68i)T^{2} \) |
| 31 | \( 1 + (-2.22 + 9.37i)T + (-27.7 - 13.9i)T^{2} \) |
| 37 | \( 1 + (7.26 + 1.28i)T + (34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (5.85 - 4.35i)T + (11.7 - 39.2i)T^{2} \) |
| 43 | \( 1 + (-0.465 - 7.98i)T + (-42.7 + 4.99i)T^{2} \) |
| 47 | \( 1 + (5.66 - 1.34i)T + (42.0 - 21.0i)T^{2} \) |
| 53 | \( 1 + (5.42 - 9.39i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.20 - 10.3i)T + (-35.2 + 47.3i)T^{2} \) |
| 61 | \( 1 + (-3.00 - 0.900i)T + (50.9 + 33.5i)T^{2} \) |
| 67 | \( 1 + (8.96 - 1.04i)T + (65.1 - 15.4i)T^{2} \) |
| 71 | \( 1 + (-9.37 + 3.41i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (2.48 + 0.903i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-11.1 - 8.30i)T + (22.6 + 75.6i)T^{2} \) |
| 83 | \( 1 + (0.216 + 0.161i)T + (23.8 + 79.5i)T^{2} \) |
| 89 | \( 1 + (2.95 - 8.12i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (1.70 - 1.11i)T + (38.4 - 89.0i)T^{2} \) |
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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
−10.88454432571617289706929701576, −9.858490012692409134885052388385, −9.352832968333086444412764819203, −8.697174771676841257637531810325, −7.918827196452939847536770759009, −6.56671784127749308567721338981, −5.88261466171847875093470905114, −4.90761742716574533097650942295, −3.84205417874477204456929367522, −2.61572698107789697606257081180,
0.22242113488209676319056104738, 1.64747745090451341251347633201, 3.27206699301372868429157336423, 3.55661492715809597005763200915, 5.21324463599941477655517112199, 6.91580562930306750209762094943, 7.13785800236534366018437118740, 8.261214801943796304424997440676, 9.258942158182372012260964416078, 9.885360476261825886364158901701
