| L(s) = 1 | − 0.628i·2-s + 1.60·4-s + 4.14i·5-s − 2.26i·8-s + 2.60·10-s − 5.40i·11-s − 3.60·13-s + 1.78·16-s + 6.66i·20-s − 3.39·22-s − 12.2·25-s + 2.26i·26-s − 5.65i·32-s + 9.39·40-s − 1.63i·41-s + ⋯ |
| L(s) = 1 | − 0.444i·2-s + 0.802·4-s + 1.85i·5-s − 0.800i·8-s + 0.823·10-s − 1.62i·11-s − 1.00·13-s + 0.447·16-s + 1.48i·20-s − 0.723·22-s − 2.44·25-s + 0.444i·26-s − 0.999i·32-s + 1.48·40-s − 0.255i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.19871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19871\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + 3.60T \) |
| good | 2 | \( 1 + 0.628iT - 2T^{2} \) |
| 5 | \( 1 - 4.14iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 5.40iT - 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 1.63iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 13.7iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 11.1iT - 59T^{2} \) |
| 61 | \( 1 + 7.21T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 7.91iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 0.380iT - 83T^{2} \) |
| 89 | \( 1 - 9.93iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−13.69731524488339160670343967695, −12.19381655291731121007740523355, −11.16981309326487115456165520414, −10.76237978891142207837728317472, −9.708756749956308101390820435147, −7.84970012995685686553693504649, −6.84444374427848748218298161413, −5.95862938509104431455822791348, −3.45648501259102381126415711926, −2.56403110260492774371489243204,
1.94411864801312061694098383356, 4.54969906997454097646307408812, 5.44489339642394308728249059601, 7.05206439937250978697192230554, 8.004433088241410228101094635496, 9.192826128835690366988213149494, 10.20875903434440903619387876426, 11.99033962088406238297244246049, 12.25900240803632837536769489476, 13.38838228684340348033758813470
