L(s) = 1 | + 2·3-s − 2·4-s − 3·5-s − 7-s + 9-s + 3·11-s − 4·12-s − 4·13-s − 6·15-s + 4·16-s − 3·17-s − 19-s + 6·20-s − 2·21-s + 4·25-s − 4·27-s + 2·28-s + 6·29-s + 4·31-s + 6·33-s + 3·35-s − 2·36-s + 2·37-s − 8·39-s + 6·41-s − 43-s − 6·44-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 4-s − 1.34·5-s − 0.377·7-s + 1/3·9-s + 0.904·11-s − 1.15·12-s − 1.10·13-s − 1.54·15-s + 16-s − 0.727·17-s − 0.229·19-s + 1.34·20-s − 0.436·21-s + 4/5·25-s − 0.769·27-s + 0.377·28-s + 1.11·29-s + 0.718·31-s + 1.04·33-s + 0.507·35-s − 1/3·36-s + 0.328·37-s − 1.28·39-s + 0.937·41-s − 0.152·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53371 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 19 | \( 1 + T \) |
| 53 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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
−14.71699995275008, −14.20078605447620, −13.87080821078644, −13.16901185146099, −12.75937248174417, −12.20182021805872, −11.75353210603811, −11.24677251536106, −10.44153590936776, −9.844957455274208, −9.360975990118001, −8.990367509797977, −8.401020426699948, −8.027783312669297, −7.613644423072211, −6.883360494566854, −6.383670255205334, −5.489952796339843, −4.611384788504225, −4.344417081851194, −3.866811241091021, −3.114712728464183, −2.802476974326507, −1.811017001355233, −0.7498825257164323, 0,
0.7498825257164323, 1.811017001355233, 2.802476974326507, 3.114712728464183, 3.866811241091021, 4.344417081851194, 4.611384788504225, 5.489952796339843, 6.383670255205334, 6.883360494566854, 7.613644423072211, 8.027783312669297, 8.401020426699948, 8.990367509797977, 9.360975990118001, 9.844957455274208, 10.44153590936776, 11.24677251536106, 11.75353210603811, 12.20182021805872, 12.75937248174417, 13.16901185146099, 13.87080821078644, 14.20078605447620, 14.71699995275008