L(s) = 1 | + 0.193·2-s − 3-s − 1.96·4-s − 0.193·6-s − 7-s − 0.768·8-s + 9-s + 2·11-s + 1.96·12-s + 1.35·13-s − 0.193·14-s + 3.77·16-s − 3.35·17-s + 0.193·18-s + 5.35·19-s + 21-s + 0.387·22-s + 4.96·23-s + 0.768·24-s + 0.261·26-s − 27-s + 1.96·28-s + 7.92·29-s + 4.57·31-s + 2.26·32-s − 2·33-s − 0.649·34-s + ⋯ |
L(s) = 1 | + 0.137·2-s − 0.577·3-s − 0.981·4-s − 0.0791·6-s − 0.377·7-s − 0.271·8-s + 0.333·9-s + 0.603·11-s + 0.566·12-s + 0.374·13-s − 0.0518·14-s + 0.943·16-s − 0.812·17-s + 0.0457·18-s + 1.22·19-s + 0.218·21-s + 0.0826·22-s + 1.03·23-s + 0.156·24-s + 0.0513·26-s − 0.192·27-s + 0.370·28-s + 1.47·29-s + 0.821·31-s + 0.401·32-s − 0.348·33-s − 0.111·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9971434834\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9971434834\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
good | 2 | \( 1 - 0.193T + 2T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 17 | \( 1 + 3.35T + 17T^{2} \) |
| 19 | \( 1 - 5.35T + 19T^{2} \) |
| 23 | \( 1 - 4.96T + 23T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 - 4.57T + 31T^{2} \) |
| 37 | \( 1 + 0.775T + 37T^{2} \) |
| 41 | \( 1 - 3.73T + 41T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 - 9.92T + 47T^{2} \) |
| 53 | \( 1 - 8.57T + 53T^{2} \) |
| 59 | \( 1 + 8.62T + 59T^{2} \) |
| 61 | \( 1 + 8.70T + 61T^{2} \) |
| 67 | \( 1 - 9.92T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 9.35T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 3.22T + 83T^{2} \) |
| 89 | \( 1 - 1.03T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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
−10.83744424866424693015182860320, −9.914525071408972659806251594373, −9.156825295931349904739131572860, −8.362766972126603023432502689627, −7.06666112289600058325462250884, −6.18421278721861225303735663511, −5.14085153600714544753454493451, −4.30027321369541196988573082776, −3.14182764210480382492854285569, −0.956151394633825783233879657697,
0.956151394633825783233879657697, 3.14182764210480382492854285569, 4.30027321369541196988573082776, 5.14085153600714544753454493451, 6.18421278721861225303735663511, 7.06666112289600058325462250884, 8.362766972126603023432502689627, 9.156825295931349904739131572860, 9.914525071408972659806251594373, 10.83744424866424693015182860320