L(s) = 1 | − 3-s + 5-s + 9-s + 4·11-s + 6·13-s − 15-s + 6·17-s − 4·19-s + 8·23-s + 25-s − 27-s − 10·29-s + 4·31-s − 4·33-s + 6·37-s − 6·39-s − 6·41-s + 4·43-s + 45-s − 12·47-s − 6·51-s − 6·53-s + 4·55-s + 4·57-s − 4·59-s − 2·61-s + 6·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 0.258·15-s + 1.45·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s + 0.718·31-s − 0.696·33-s + 0.986·37-s − 0.960·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s − 1.75·47-s − 0.840·51-s − 0.824·53-s + 0.539·55-s + 0.529·57-s − 0.520·59-s − 0.256·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.058761654\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.058761654\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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
−14.64379037632096, −14.07012801637751, −13.51624298299225, −12.92430268869563, −12.72132775572923, −11.91716594203712, −11.42555827642020, −10.97813720464688, −10.65817502260631, −9.771109249408326, −9.453469581624895, −8.906576042011783, −8.282448626061115, −7.739956820021511, −6.905584371387554, −6.479617911021403, −6.037241982381170, −5.506445369136337, −4.853432334854833, −4.129460896559364, −3.524000927079925, −3.050972773050760, −1.818260327497605, −1.361830678912081, −0.7131135993202221,
0.7131135993202221, 1.361830678912081, 1.818260327497605, 3.050972773050760, 3.524000927079925, 4.129460896559364, 4.853432334854833, 5.506445369136337, 6.037241982381170, 6.479617911021403, 6.905584371387554, 7.739956820021511, 8.282448626061115, 8.906576042011783, 9.453469581624895, 9.771109249408326, 10.65817502260631, 10.97813720464688, 11.42555827642020, 11.91716594203712, 12.72132775572923, 12.92430268869563, 13.51624298299225, 14.07012801637751, 14.64379037632096