L(s) = 1 | + 1.47·3-s − 5-s − 3.14·7-s − 0.812·9-s + 1.94·11-s − 6.40·13-s − 1.47·15-s − 6.69·17-s + 6.85·19-s − 4.65·21-s − 4.16·23-s + 25-s − 5.63·27-s + 4.23·29-s − 8.26·31-s + 2.88·33-s + 3.14·35-s + 4.63·37-s − 9.46·39-s − 5.88·41-s + 43-s + 0.812·45-s + 1.22·47-s + 2.88·49-s − 9.89·51-s − 1.49·53-s − 1.94·55-s + ⋯ |
L(s) = 1 | + 0.853·3-s − 0.447·5-s − 1.18·7-s − 0.270·9-s + 0.587·11-s − 1.77·13-s − 0.381·15-s − 1.62·17-s + 1.57·19-s − 1.01·21-s − 0.867·23-s + 0.200·25-s − 1.08·27-s + 0.787·29-s − 1.48·31-s + 0.501·33-s + 0.531·35-s + 0.762·37-s − 1.51·39-s − 0.918·41-s + 0.152·43-s + 0.121·45-s + 0.179·47-s + 0.412·49-s − 1.38·51-s − 0.205·53-s − 0.262·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 - T \) |
good | 3 | \( 1 - 1.47T + 3T^{2} \) |
| 7 | \( 1 + 3.14T + 7T^{2} \) |
| 11 | \( 1 - 1.94T + 11T^{2} \) |
| 13 | \( 1 + 6.40T + 13T^{2} \) |
| 17 | \( 1 + 6.69T + 17T^{2} \) |
| 19 | \( 1 - 6.85T + 19T^{2} \) |
| 23 | \( 1 + 4.16T + 23T^{2} \) |
| 29 | \( 1 - 4.23T + 29T^{2} \) |
| 31 | \( 1 + 8.26T + 31T^{2} \) |
| 37 | \( 1 - 4.63T + 37T^{2} \) |
| 41 | \( 1 + 5.88T + 41T^{2} \) |
| 47 | \( 1 - 1.22T + 47T^{2} \) |
| 53 | \( 1 + 1.49T + 53T^{2} \) |
| 59 | \( 1 - 7.76T + 59T^{2} \) |
| 61 | \( 1 - 3.57T + 61T^{2} \) |
| 67 | \( 1 + 8.72T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 0.214T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 5.91T + 97T^{2} \) |
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
−9.378116415068003308199577013874, −9.228714182209691545517741497074, −8.034133299506803719300487479957, −7.24272113350508853252817636904, −6.49639786426084099683488052638, −5.25443300458815023596763753550, −4.07152488398894387279425890795, −3.14843146260007048887222979845, −2.29403700969837752233679801011, 0,
2.29403700969837752233679801011, 3.14843146260007048887222979845, 4.07152488398894387279425890795, 5.25443300458815023596763753550, 6.49639786426084099683488052638, 7.24272113350508853252817636904, 8.034133299506803719300487479957, 9.228714182209691545517741497074, 9.378116415068003308199577013874