L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s + 2·11-s − 12-s + 7·13-s + 15-s + 16-s − 18-s + 19-s − 20-s − 2·22-s + 23-s + 24-s + 25-s − 7·26-s − 27-s + 4·29-s − 30-s + 9·31-s − 32-s − 2·33-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s − 0.288·12-s + 1.94·13-s + 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.229·19-s − 0.223·20-s − 0.426·22-s + 0.208·23-s + 0.204·24-s + 1/5·25-s − 1.37·26-s − 0.192·27-s + 0.742·29-s − 0.182·30-s + 1.61·31-s − 0.176·32-s − 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.765225740\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.765225740\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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
−13.17586359662437, −12.73741463809377, −11.97628694948411, −11.78352466072073, −11.39634638400713, −10.84126963195194, −10.40713360359915, −10.09331073859183, −9.321490988905248, −8.859787056132727, −8.527311038981639, −8.019148392217881, −7.468130806262806, −6.814981447759704, −6.546048954088036, −5.824136570084820, −5.729337464438387, −4.651688709114551, −4.292986630319048, −3.694816883742165, −3.089505797212364, −2.491488143233143, −1.424696329176720, −1.172076578145796, −0.5274372098374442,
0.5274372098374442, 1.172076578145796, 1.424696329176720, 2.491488143233143, 3.089505797212364, 3.694816883742165, 4.292986630319048, 4.651688709114551, 5.729337464438387, 5.824136570084820, 6.546048954088036, 6.814981447759704, 7.468130806262806, 8.019148392217881, 8.527311038981639, 8.859787056132727, 9.321490988905248, 10.09331073859183, 10.40713360359915, 10.84126963195194, 11.39634638400713, 11.78352466072073, 11.97628694948411, 12.73741463809377, 13.17586359662437