L(s) = 1 | − 5-s − 4·7-s − 4·11-s − 2·13-s − 2·17-s + 6·19-s − 4·25-s + 29-s − 3·31-s + 4·35-s − 4·37-s − 6·41-s − 4·43-s − 2·47-s + 9·49-s + 6·53-s + 4·55-s + 5·59-s + 2·61-s + 2·65-s + 8·67-s − 8·71-s + 6·73-s + 16·77-s − 3·79-s − 3·83-s + 2·85-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.51·7-s − 1.20·11-s − 0.554·13-s − 0.485·17-s + 1.37·19-s − 4/5·25-s + 0.185·29-s − 0.538·31-s + 0.676·35-s − 0.657·37-s − 0.937·41-s − 0.609·43-s − 0.291·47-s + 9/7·49-s + 0.824·53-s + 0.539·55-s + 0.650·59-s + 0.256·61-s + 0.248·65-s + 0.977·67-s − 0.949·71-s + 0.702·73-s + 1.82·77-s − 0.337·79-s − 0.329·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−13.71269998252114, −13.25342725338537, −13.03870085238598, −12.39391238200084, −11.94949342176626, −11.57609701419246, −10.88733646916012, −10.34200726425509, −9.867108158858052, −9.663609987446944, −8.972776952366197, −8.429058399628008, −7.848674135089500, −7.330134019993642, −6.960997985656161, −6.411141665681185, −5.735178838551004, −5.227243875604467, −4.847039187025473, −3.804798703812430, −3.626220660782005, −2.861033107695311, −2.507292109753956, −1.641165992121635, −0.5509713777073990, 0,
0.5509713777073990, 1.641165992121635, 2.507292109753956, 2.861033107695311, 3.626220660782005, 3.804798703812430, 4.847039187025473, 5.227243875604467, 5.735178838551004, 6.411141665681185, 6.960997985656161, 7.330134019993642, 7.848674135089500, 8.429058399628008, 8.972776952366197, 9.663609987446944, 9.867108158858052, 10.34200726425509, 10.88733646916012, 11.57609701419246, 11.94949342176626, 12.39391238200084, 13.03870085238598, 13.25342725338537, 13.71269998252114