| L(s) = 1 | + 3-s − 5-s − 7-s − 2·9-s − 6·11-s − 3·13-s − 15-s + 2·17-s + 7·19-s − 21-s + 3·23-s − 4·25-s − 5·27-s + 4·31-s − 6·33-s + 35-s + 4·37-s − 3·39-s − 12·41-s + 2·45-s + 6·47-s + 49-s + 2·51-s − 12·53-s + 6·55-s + 7·57-s + 5·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s − 1.80·11-s − 0.832·13-s − 0.258·15-s + 0.485·17-s + 1.60·19-s − 0.218·21-s + 0.625·23-s − 4/5·25-s − 0.962·27-s + 0.718·31-s − 1.04·33-s + 0.169·35-s + 0.657·37-s − 0.480·39-s − 1.87·41-s + 0.298·45-s + 0.875·47-s + 1/7·49-s + 0.280·51-s − 1.64·53-s + 0.809·55-s + 0.927·57-s + 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−14.94370970343000, −14.23749528944227, −13.79420746721551, −13.44573186372413, −12.84261477117489, −12.24834044692165, −11.82955478319465, −11.26032532069227, −10.69422848021728, −10.02209005089119, −9.661727088284015, −9.189480902652667, −8.301802594490490, −7.982316032205883, −7.608915410621466, −7.067307250706216, −6.250618183339016, −5.467187525522963, −5.192101495198611, −4.580450507164488, −3.478263295971162, −3.220468666919959, −2.628483081322078, −2.025758492566862, −0.7870963226565282, 0,
0.7870963226565282, 2.025758492566862, 2.628483081322078, 3.220468666919959, 3.478263295971162, 4.580450507164488, 5.192101495198611, 5.467187525522963, 6.250618183339016, 7.067307250706216, 7.608915410621466, 7.982316032205883, 8.301802594490490, 9.189480902652667, 9.661727088284015, 10.02209005089119, 10.69422848021728, 11.26032532069227, 11.82955478319465, 12.24834044692165, 12.84261477117489, 13.44573186372413, 13.79420746721551, 14.23749528944227, 14.94370970343000
