L(s) = 1 | + 2.31·2-s + 3.37·4-s + 2.47·5-s + 3.26·7-s + 3.18·8-s + 5.74·10-s − 11-s + 6.05·13-s + 7.57·14-s + 0.629·16-s + 6.63·17-s − 4.68·19-s + 8.36·20-s − 2.31·22-s + 6.72·23-s + 1.14·25-s + 14.0·26-s + 11.0·28-s + 6.44·29-s − 6.17·31-s − 4.90·32-s + 15.3·34-s + 8.10·35-s − 6.59·37-s − 10.8·38-s + 7.88·40-s − 8.91·41-s + ⋯ |
L(s) = 1 | + 1.63·2-s + 1.68·4-s + 1.10·5-s + 1.23·7-s + 1.12·8-s + 1.81·10-s − 0.301·11-s + 1.67·13-s + 2.02·14-s + 0.157·16-s + 1.60·17-s − 1.07·19-s + 1.86·20-s − 0.494·22-s + 1.40·23-s + 0.229·25-s + 2.75·26-s + 2.08·28-s + 1.19·29-s − 1.10·31-s − 0.866·32-s + 2.63·34-s + 1.36·35-s − 1.08·37-s − 1.76·38-s + 1.24·40-s − 1.39·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.351468420\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.351468420\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 - 2.31T + 2T^{2} \) |
| 5 | \( 1 - 2.47T + 5T^{2} \) |
| 7 | \( 1 - 3.26T + 7T^{2} \) |
| 13 | \( 1 - 6.05T + 13T^{2} \) |
| 17 | \( 1 - 6.63T + 17T^{2} \) |
| 19 | \( 1 + 4.68T + 19T^{2} \) |
| 23 | \( 1 - 6.72T + 23T^{2} \) |
| 29 | \( 1 - 6.44T + 29T^{2} \) |
| 31 | \( 1 + 6.17T + 31T^{2} \) |
| 37 | \( 1 + 6.59T + 37T^{2} \) |
| 41 | \( 1 + 8.91T + 41T^{2} \) |
| 43 | \( 1 + 8.98T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 + 3.81T + 59T^{2} \) |
| 67 | \( 1 - 3.27T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 6.29T + 73T^{2} \) |
| 79 | \( 1 + 5.89T + 79T^{2} \) |
| 83 | \( 1 - 6.77T + 83T^{2} \) |
| 89 | \( 1 - 2.86T + 89T^{2} \) |
| 97 | \( 1 - 8.49T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.141297084148019575800445696226, −6.96371941847619912476651060403, −6.38322410985918222677903866740, −5.74319102851425006933875444487, −5.08554736276790698289724758322, −4.76706278075950709325733061310, −3.55469113046909570511853946334, −3.14346096574963001617403191717, −1.85834950938306942531490538669, −1.46687867678198896744910628507,
1.46687867678198896744910628507, 1.85834950938306942531490538669, 3.14346096574963001617403191717, 3.55469113046909570511853946334, 4.76706278075950709325733061310, 5.08554736276790698289724758322, 5.74319102851425006933875444487, 6.38322410985918222677903866740, 6.96371941847619912476651060403, 8.141297084148019575800445696226