L(s) = 1 | − 5-s − 0.627·7-s + 1.35·11-s + 2.31·13-s − 5.74·17-s − 8.04·19-s − 23-s + 25-s + 6.21·29-s − 7.91·31-s + 0.627·35-s + 10.0·37-s + 2.48·41-s + 4.42·43-s + 4.68·47-s − 6.60·49-s − 2.21·53-s − 1.35·55-s + 0.999·59-s − 9.30·61-s − 2.31·65-s − 11.0·67-s + 9.42·71-s + 9.73·73-s − 0.847·77-s + 3.26·79-s − 1.04·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.237·7-s + 0.407·11-s + 0.643·13-s − 1.39·17-s − 1.84·19-s − 0.208·23-s + 0.200·25-s + 1.15·29-s − 1.42·31-s + 0.106·35-s + 1.65·37-s + 0.388·41-s + 0.674·43-s + 0.683·47-s − 0.943·49-s − 0.304·53-s − 0.182·55-s + 0.130·59-s − 1.19·61-s − 0.287·65-s − 1.34·67-s + 1.11·71-s + 1.13·73-s − 0.0965·77-s + 0.367·79-s − 0.114·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.345881099\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.345881099\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + 0.627T + 7T^{2} \) |
| 11 | \( 1 - 1.35T + 11T^{2} \) |
| 13 | \( 1 - 2.31T + 13T^{2} \) |
| 17 | \( 1 + 5.74T + 17T^{2} \) |
| 19 | \( 1 + 8.04T + 19T^{2} \) |
| 29 | \( 1 - 6.21T + 29T^{2} \) |
| 31 | \( 1 + 7.91T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 2.48T + 41T^{2} \) |
| 43 | \( 1 - 4.42T + 43T^{2} \) |
| 47 | \( 1 - 4.68T + 47T^{2} \) |
| 53 | \( 1 + 2.21T + 53T^{2} \) |
| 59 | \( 1 - 0.999T + 59T^{2} \) |
| 61 | \( 1 + 9.30T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 - 9.42T + 71T^{2} \) |
| 73 | \( 1 - 9.73T + 73T^{2} \) |
| 79 | \( 1 - 3.26T + 79T^{2} \) |
| 83 | \( 1 + 1.04T + 83T^{2} \) |
| 89 | \( 1 + 2.84T + 89T^{2} \) |
| 97 | \( 1 - 7.81T + 97T^{2} \) |
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
−7.84614321464516123928964328381, −7.06989728700488199903865055788, −6.25421031237672510058406891485, −6.11499196460167158186430716873, −4.75733493995032166478732384257, −4.28671727233118180594901520582, −3.63858982427455555017185449395, −2.61337326127592544255698879410, −1.83640526174772949237104591922, −0.55705924827714916080852002623,
0.55705924827714916080852002623, 1.83640526174772949237104591922, 2.61337326127592544255698879410, 3.63858982427455555017185449395, 4.28671727233118180594901520582, 4.75733493995032166478732384257, 6.11499196460167158186430716873, 6.25421031237672510058406891485, 7.06989728700488199903865055788, 7.84614321464516123928964328381