| L(s) = 1 | + (−0.742 + 0.229i)2-s + (−0.365 − 0.930i)3-s + (−1.15 + 0.786i)4-s + (1.96 + 0.295i)5-s + (0.484 + 0.607i)6-s + (1.38 − 2.25i)7-s + (1.64 − 2.06i)8-s + (−0.733 + 0.680i)9-s + (−1.52 + 0.229i)10-s + (4.54 + 4.21i)11-s + (1.15 + 0.786i)12-s + (1.11 − 4.87i)13-s + (−0.508 + 1.99i)14-s + (−0.441 − 1.93i)15-s + (0.269 − 0.687i)16-s + (0.0440 − 0.588i)17-s + ⋯ |
| L(s) = 1 | + (−0.525 + 0.162i)2-s + (−0.210 − 0.537i)3-s + (−0.576 + 0.393i)4-s + (0.876 + 0.132i)5-s + (0.197 + 0.248i)6-s + (0.521 − 0.853i)7-s + (0.581 − 0.729i)8-s + (−0.244 + 0.226i)9-s + (−0.481 + 0.0726i)10-s + (1.37 + 1.27i)11-s + (0.332 + 0.226i)12-s + (0.308 − 1.35i)13-s + (−0.135 + 0.532i)14-s + (−0.113 − 0.499i)15-s + (0.0674 − 0.171i)16-s + (0.0106 − 0.142i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.240i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.874106 - 0.106648i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.874106 - 0.106648i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.365 + 0.930i)T \) |
| 7 | \( 1 + (-1.38 + 2.25i)T \) |
| good | 2 | \( 1 + (0.742 - 0.229i)T + (1.65 - 1.12i)T^{2} \) |
| 5 | \( 1 + (-1.96 - 0.295i)T + (4.77 + 1.47i)T^{2} \) |
| 11 | \( 1 + (-4.54 - 4.21i)T + (0.822 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-1.11 + 4.87i)T + (-11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-0.0440 + 0.588i)T + (-16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (-0.327 - 0.566i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.115 - 1.53i)T + (-22.7 + 3.42i)T^{2} \) |
| 29 | \( 1 + (4.13 - 1.98i)T + (18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (4.14 - 7.17i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.95 + 4.74i)T + (13.5 + 34.4i)T^{2} \) |
| 41 | \( 1 + (-4.92 + 6.17i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (-2.05 - 2.57i)T + (-9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (1.79 - 0.553i)T + (38.8 - 26.4i)T^{2} \) |
| 53 | \( 1 + (5.45 - 3.71i)T + (19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (-4.21 + 0.634i)T + (56.3 - 17.3i)T^{2} \) |
| 61 | \( 1 + (-11.9 - 8.11i)T + (22.2 + 56.7i)T^{2} \) |
| 67 | \( 1 + (-1.01 + 1.75i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.23 + 3.00i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (6.44 + 1.98i)T + (60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + (-0.105 - 0.183i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.38 - 6.08i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (8.47 - 7.86i)T + (6.65 - 88.7i)T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−13.01200834604983822795397496827, −12.24512106567223341310340136319, −10.76453164455726423244317938939, −9.872170670191987179482796924641, −8.903967082098816121654081401284, −7.61984051923499089437859689027, −6.91603017708327669376583261365, −5.35558843626018514925610664117, −3.87297954322701044196572016444, −1.45423139095334761798739781306,
1.72318222888563472641143873419, 4.10048270637205235443508996525, 5.47382122464442972240282678422, 6.27496149110044000446410012353, 8.453418830776445955208977601175, 9.153639362917087800349787830599, 9.696460504703170942952812834625, 11.16057497844662448420686974015, 11.61598003493203438666100375271, 13.31395249771656919099770167280
