| L(s) = 1 | − 0.704·2-s + 3-s − 1.50·4-s − 1.50·5-s − 0.704·6-s + 2.46·8-s + 9-s + 1.05·10-s − 4.26·11-s − 1.50·12-s − 2.30·13-s − 1.50·15-s + 1.26·16-s − 0.905·17-s − 0.704·18-s − 1.17·19-s + 2.26·20-s + 3.00·22-s − 0.893·23-s + 2.46·24-s − 2.73·25-s + 1.62·26-s + 27-s + 3.32·29-s + 1.05·30-s + 3.09·31-s − 5.83·32-s + ⋯ |
| L(s) = 1 | − 0.498·2-s + 0.577·3-s − 0.751·4-s − 0.672·5-s − 0.287·6-s + 0.872·8-s + 0.333·9-s + 0.335·10-s − 1.28·11-s − 0.434·12-s − 0.638·13-s − 0.388·15-s + 0.316·16-s − 0.219·17-s − 0.166·18-s − 0.269·19-s + 0.505·20-s + 0.641·22-s − 0.186·23-s + 0.503·24-s − 0.547·25-s + 0.318·26-s + 0.192·27-s + 0.617·29-s + 0.193·30-s + 0.555·31-s − 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6642630577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6642630577\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 0.704T + 2T^{2} \) |
| 5 | \( 1 + 1.50T + 5T^{2} \) |
| 11 | \( 1 + 4.26T + 11T^{2} \) |
| 13 | \( 1 + 2.30T + 13T^{2} \) |
| 17 | \( 1 + 0.905T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 + 0.893T + 23T^{2} \) |
| 29 | \( 1 - 3.32T + 29T^{2} \) |
| 31 | \( 1 - 3.09T + 31T^{2} \) |
| 37 | \( 1 + 2.36T + 37T^{2} \) |
| 43 | \( 1 + 7.84T + 43T^{2} \) |
| 47 | \( 1 - 1.30T + 47T^{2} \) |
| 53 | \( 1 + 6.64T + 53T^{2} \) |
| 59 | \( 1 - 0.638T + 59T^{2} \) |
| 61 | \( 1 - 9.84T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 0.245T + 71T^{2} \) |
| 73 | \( 1 - 2.08T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 6.41T + 83T^{2} \) |
| 89 | \( 1 - 1.81T + 89T^{2} \) |
| 97 | \( 1 - 6.29T + 97T^{2} \) |
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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
−8.205587441701624177590607663774, −7.60645603274251702812783199026, −7.03797192964273824858895415357, −5.89869234828497478361617429720, −4.93254017753831758949180927288, −4.51917019306950125394227445550, −3.62742135017520801864893846074, −2.79849874987979233633920068320, −1.82293284791531446133331586147, −0.43968734951597055735551428549,
0.43968734951597055735551428549, 1.82293284791531446133331586147, 2.79849874987979233633920068320, 3.62742135017520801864893846074, 4.51917019306950125394227445550, 4.93254017753831758949180927288, 5.89869234828497478361617429720, 7.03797192964273824858895415357, 7.60645603274251702812783199026, 8.205587441701624177590607663774
