L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 3·8-s + 9-s − 5·11-s − 12-s + 13-s − 16-s + 7·17-s − 18-s − 19-s + 5·22-s + 8·23-s + 3·24-s − 26-s + 27-s − 6·29-s + 31-s − 5·32-s − 5·33-s − 7·34-s − 36-s − 10·37-s + 38-s + 39-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 1.50·11-s − 0.288·12-s + 0.277·13-s − 1/4·16-s + 1.69·17-s − 0.235·18-s − 0.229·19-s + 1.06·22-s + 1.66·23-s + 0.612·24-s − 0.196·26-s + 0.192·27-s − 1.11·29-s + 0.179·31-s − 0.883·32-s − 0.870·33-s − 1.20·34-s − 1/6·36-s − 1.64·37-s + 0.162·38-s + 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.710722837\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.710722837\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.66782192698783, −13.98030859016357, −13.53973235320456, −13.14321181804882, −12.52136221056088, −12.28691783861504, −11.09508361363253, −10.89221239902048, −10.25755750857653, −9.890878886185966, −9.243693654406070, −8.880078102033718, −8.152138725005980, −7.953976923098693, −7.246081583965584, −6.988703697076092, −5.714970619919396, −5.384458305667272, −4.897002304854212, −4.001612717287882, −3.501837506916322, −2.789096519113255, −2.106626176138019, −1.182949782340153, −0.5760127332362205,
0.5760127332362205, 1.182949782340153, 2.106626176138019, 2.789096519113255, 3.501837506916322, 4.001612717287882, 4.897002304854212, 5.384458305667272, 5.714970619919396, 6.988703697076092, 7.246081583965584, 7.953976923098693, 8.152138725005980, 8.880078102033718, 9.243693654406070, 9.890878886185966, 10.25755750857653, 10.89221239902048, 11.09508361363253, 12.28691783861504, 12.52136221056088, 13.14321181804882, 13.53973235320456, 13.98030859016357, 14.66782192698783