L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 3·7-s − 8-s − 2·9-s − 10-s − 2·11-s − 12-s − 5·13-s + 3·14-s − 15-s + 16-s + 17-s + 2·18-s − 19-s + 20-s + 3·21-s + 2·22-s − 4·23-s + 24-s + 25-s + 5·26-s + 5·27-s − 3·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.603·11-s − 0.288·12-s − 1.38·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.471·18-s − 0.229·19-s + 0.223·20-s + 0.654·21-s + 0.426·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.980·26-s + 0.962·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 139 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 17 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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
−15.44616759825014, −14.77723379798389, −14.38643073906321, −13.61131647258587, −13.24073115913288, −12.45664890032581, −12.17161744019683, −11.69648370088138, −11.02364714578287, −10.35328446049797, −9.989577058393511, −9.732297339500303, −9.002739382785286, −8.404056261732485, −7.868046051395565, −7.160799772298750, −6.689218301185839, −5.989254774698292, −5.785054631820648, −4.946061640972337, −4.386004439245018, −3.230613532242011, −2.797909435277998, −2.265719771203788, −1.227155572551433, 0, 0,
1.227155572551433, 2.265719771203788, 2.797909435277998, 3.230613532242011, 4.386004439245018, 4.946061640972337, 5.785054631820648, 5.989254774698292, 6.689218301185839, 7.160799772298750, 7.868046051395565, 8.404056261732485, 9.002739382785286, 9.732297339500303, 9.989577058393511, 10.35328446049797, 11.02364714578287, 11.69648370088138, 12.17161744019683, 12.45664890032581, 13.24073115913288, 13.61131647258587, 14.38643073906321, 14.77723379798389, 15.44616759825014