L(s) = 1 | − 1.94·2-s + 3.14·3-s + 1.77·4-s + 5-s − 6.11·6-s + 7-s + 0.427·8-s + 6.87·9-s − 1.94·10-s + 5.04·11-s + 5.59·12-s + 5.16·13-s − 1.94·14-s + 3.14·15-s − 4.39·16-s − 7.40·17-s − 13.3·18-s − 5.95·19-s + 1.77·20-s + 3.14·21-s − 9.80·22-s + 23-s + 1.34·24-s + 25-s − 10.0·26-s + 12.1·27-s + 1.77·28-s + ⋯ |
L(s) = 1 | − 1.37·2-s + 1.81·3-s + 0.889·4-s + 0.447·5-s − 2.49·6-s + 0.377·7-s + 0.151·8-s + 2.29·9-s − 0.614·10-s + 1.52·11-s + 1.61·12-s + 1.43·13-s − 0.519·14-s + 0.811·15-s − 1.09·16-s − 1.79·17-s − 3.15·18-s − 1.36·19-s + 0.397·20-s + 0.685·21-s − 2.09·22-s + 0.208·23-s + 0.274·24-s + 0.200·25-s − 1.96·26-s + 2.34·27-s + 0.336·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.757578533\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.757578533\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + 1.94T + 2T^{2} \) |
| 3 | \( 1 - 3.14T + 3T^{2} \) |
| 11 | \( 1 - 5.04T + 11T^{2} \) |
| 13 | \( 1 - 5.16T + 13T^{2} \) |
| 17 | \( 1 + 7.40T + 17T^{2} \) |
| 19 | \( 1 + 5.95T + 19T^{2} \) |
| 29 | \( 1 + 0.530T + 29T^{2} \) |
| 31 | \( 1 + 2.50T + 31T^{2} \) |
| 37 | \( 1 + 7.70T + 37T^{2} \) |
| 41 | \( 1 + 2.51T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 0.447T + 47T^{2} \) |
| 53 | \( 1 - 3.18T + 53T^{2} \) |
| 59 | \( 1 - 9.09T + 59T^{2} \) |
| 61 | \( 1 + 0.608T + 61T^{2} \) |
| 67 | \( 1 - 8.89T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 2.19T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 8.82T + 89T^{2} \) |
| 97 | \( 1 - 0.311T + 97T^{2} \) |
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
−9.912765182676904261764635154955, −8.901521771088979062209907850225, −8.799181050735022834080785868505, −8.314505769012276061884596976638, −6.99075530524209567057914509859, −6.55236900457127908900002672996, −4.44098673975622595296140222765, −3.65629896819087111291199195404, −2.09007615117882363262516400958, −1.53224919879341955957944570291,
1.53224919879341955957944570291, 2.09007615117882363262516400958, 3.65629896819087111291199195404, 4.44098673975622595296140222765, 6.55236900457127908900002672996, 6.99075530524209567057914509859, 8.314505769012276061884596976638, 8.799181050735022834080785868505, 8.901521771088979062209907850225, 9.912765182676904261764635154955