| L(s) = 1 | + (0.489 − 1.32i)2-s + (1.72 + 0.122i)3-s + (−1.51 − 1.29i)4-s + (2.35 + 1.75i)5-s + (1.00 − 2.23i)6-s + (0.679 − 1.03i)7-s + (−2.46 + 1.37i)8-s + (2.97 + 0.422i)9-s + (3.48 − 2.26i)10-s + (−2.82 + 0.329i)11-s + (−2.46 − 2.43i)12-s + (−0.150 + 0.501i)13-s + (−1.03 − 1.40i)14-s + (3.85 + 3.31i)15-s + (0.620 + 3.95i)16-s + (1.84 − 0.325i)17-s + ⋯ |
| L(s) = 1 | + (0.346 − 0.938i)2-s + (0.997 + 0.0705i)3-s + (−0.759 − 0.649i)4-s + (1.05 + 0.784i)5-s + (0.411 − 0.911i)6-s + (0.256 − 0.390i)7-s + (−0.873 + 0.487i)8-s + (0.990 + 0.140i)9-s + (1.10 − 0.716i)10-s + (−0.850 + 0.0993i)11-s + (−0.712 − 0.701i)12-s + (−0.0416 + 0.139i)13-s + (−0.277 − 0.376i)14-s + (0.995 + 0.856i)15-s + (0.155 + 0.987i)16-s + (0.448 − 0.0790i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.93071 - 1.06659i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.93071 - 1.06659i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.489 + 1.32i)T \) |
| 3 | \( 1 + (-1.72 - 0.122i)T \) |
| good | 5 | \( 1 + (-2.35 - 1.75i)T + (1.43 + 4.78i)T^{2} \) |
| 7 | \( 1 + (-0.679 + 1.03i)T + (-2.77 - 6.42i)T^{2} \) |
| 11 | \( 1 + (2.82 - 0.329i)T + (10.7 - 2.53i)T^{2} \) |
| 13 | \( 1 + (0.150 - 0.501i)T + (-10.8 - 7.14i)T^{2} \) |
| 17 | \( 1 + (-1.84 + 0.325i)T + (15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (5.54 + 0.978i)T + (17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (-2.77 + 1.82i)T + (9.10 - 21.1i)T^{2} \) |
| 29 | \( 1 + (5.69 + 5.37i)T + (1.68 + 28.9i)T^{2} \) |
| 31 | \( 1 + (8.18 - 0.476i)T + (30.7 - 3.59i)T^{2} \) |
| 37 | \( 1 + (-0.151 - 0.0551i)T + (28.3 + 23.7i)T^{2} \) |
| 41 | \( 1 + (-2.08 - 8.81i)T + (-36.6 + 18.4i)T^{2} \) |
| 43 | \( 1 + (7.79 + 3.36i)T + (29.5 + 31.2i)T^{2} \) |
| 47 | \( 1 + (-0.0268 + 0.461i)T + (-46.6 - 5.45i)T^{2} \) |
| 53 | \( 1 + (-0.908 + 0.524i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-10.0 - 1.16i)T + (57.4 + 13.6i)T^{2} \) |
| 61 | \( 1 + (5.86 - 2.94i)T + (36.4 - 48.9i)T^{2} \) |
| 67 | \( 1 + (-6.87 + 6.48i)T + (3.89 - 66.8i)T^{2} \) |
| 71 | \( 1 + (1.57 - 1.32i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-1.15 - 0.970i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (3.45 - 14.5i)T + (-70.5 - 35.4i)T^{2} \) |
| 83 | \( 1 + (-12.9 - 3.07i)T + (74.1 + 37.2i)T^{2} \) |
| 89 | \( 1 + (-8.43 + 10.0i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (1.25 + 1.68i)T + (-27.8 + 92.9i)T^{2} \) |
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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
−11.18187638684521163439757195336, −10.45865780368600771358769461163, −9.843034477598838665333383191421, −8.961069958083798839854582185335, −7.80954302525785304660904760899, −6.54029521426486135578695553194, −5.24757758340115509883510119715, −3.99649107045902317548968224191, −2.75165305142764109844056261250, −1.92958509434008083967671876240,
2.03950409985197947702903007056, 3.58374250341385934185789712527, 5.02003823513426964479183933344, 5.69150359409108285914731146755, 7.04341247292866627213255999799, 8.053153608821620996828691805939, 8.849929208839142284837618426691, 9.419206734806578445221853773570, 10.55098960440456553326627773571, 12.33969748674920408651315407078
