| L(s) = 1 | + (−0.887 − 1.10i)2-s + (0.547 + 1.64i)3-s + (−0.423 + 1.95i)4-s + (0.281 − 0.604i)5-s + (1.32 − 2.06i)6-s + (1.65 − 0.292i)7-s + (2.52 − 1.26i)8-s + (−2.40 + 1.79i)9-s + (−0.915 + 0.226i)10-s + (3.35 − 1.56i)11-s + (−3.44 + 0.374i)12-s + (0.100 + 1.14i)13-s + (−1.79 − 1.56i)14-s + (1.14 + 0.132i)15-s + (−3.64 − 1.65i)16-s + (0.0294 − 0.0510i)17-s + ⋯ |
| L(s) = 1 | + (−0.627 − 0.778i)2-s + (0.316 + 0.948i)3-s + (−0.211 + 0.977i)4-s + (0.125 − 0.270i)5-s + (0.539 − 0.841i)6-s + (0.626 − 0.110i)7-s + (0.893 − 0.448i)8-s + (−0.800 + 0.599i)9-s + (−0.289 + 0.0715i)10-s + (1.01 − 0.471i)11-s + (−0.994 + 0.108i)12-s + (0.0277 + 0.317i)13-s + (−0.479 − 0.418i)14-s + (0.296 + 0.0340i)15-s + (−0.910 − 0.413i)16-s + (0.00715 − 0.0123i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21351 + 0.177911i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21351 + 0.177911i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.887 + 1.10i)T \) |
| 3 | \( 1 + (-0.547 - 1.64i)T \) |
| good | 5 | \( 1 + (-0.281 + 0.604i)T + (-3.21 - 3.83i)T^{2} \) |
| 7 | \( 1 + (-1.65 + 0.292i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-3.35 + 1.56i)T + (7.07 - 8.42i)T^{2} \) |
| 13 | \( 1 + (-0.100 - 1.14i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-0.0294 + 0.0510i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.22 - 4.58i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-2.16 - 0.381i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-7.00 - 0.612i)T + (28.5 + 5.03i)T^{2} \) |
| 31 | \( 1 + (-0.525 + 2.98i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.28 - 8.53i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (2.54 - 3.03i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (2.78 - 1.29i)T + (27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (0.554 + 3.14i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-3.08 + 3.08i)T - 53iT^{2} \) |
| 59 | \( 1 + (-3.89 + 8.35i)T + (-37.9 - 45.1i)T^{2} \) |
| 61 | \( 1 + (-7.20 + 10.2i)T + (-20.8 - 57.3i)T^{2} \) |
| 67 | \( 1 + (0.541 + 6.18i)T + (-65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (3.99 + 2.30i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.34 + 1.35i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.20 - 6.88i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (11.1 + 0.972i)T + (81.7 + 14.4i)T^{2} \) |
| 89 | \( 1 + (13.5 - 7.85i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.17 + 2.61i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.23985137320286536404122969884, −10.15296783983758273915257552639, −9.537408654506915409352699831124, −8.558369996579935171818199899897, −8.142155904961070038479719149876, −6.61792257043158971856229892211, −5.03795601534774315733964928950, −4.11176889303726241993131174387, −3.10767151850915229784827133795, −1.51310034625540432146811144442,
1.12150938298910920024265393710, 2.50800139150993749877341512786, 4.48606882397889828094731828537, 5.75069057841718691714587645249, 6.80021158788986495739681900545, 7.22470646029413174915862633931, 8.488515504346164838674284292725, 8.847904699542703981020535106944, 10.03911179070559903179293949131, 11.06685364086133262225325216206
