L(s) = 1 | + 8.92·3-s − 11.8·5-s + 9.85·7-s + 52.7·9-s + 39.0·11-s + 91.5·13-s − 105.·15-s − 37.1·17-s − 46.4·19-s + 87.9·21-s − 120.·23-s + 15.5·25-s + 229.·27-s − 27.2·29-s − 81.1·31-s + 348.·33-s − 116.·35-s + 10.9·37-s + 817.·39-s − 205.·41-s + 115.·43-s − 624.·45-s − 312.·47-s − 245.·49-s − 331.·51-s + 90.9·53-s − 463.·55-s + ⋯ |
L(s) = 1 | + 1.71·3-s − 1.06·5-s + 0.532·7-s + 1.95·9-s + 1.07·11-s + 1.95·13-s − 1.82·15-s − 0.529·17-s − 0.561·19-s + 0.914·21-s − 1.09·23-s + 0.124·25-s + 1.63·27-s − 0.174·29-s − 0.470·31-s + 1.84·33-s − 0.564·35-s + 0.0488·37-s + 3.35·39-s − 0.782·41-s + 0.409·43-s − 2.07·45-s − 0.970·47-s − 0.716·49-s − 0.910·51-s + 0.235·53-s − 1.13·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.669190455\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.669190455\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 - 8.92T + 27T^{2} \) |
| 5 | \( 1 + 11.8T + 125T^{2} \) |
| 7 | \( 1 - 9.85T + 343T^{2} \) |
| 11 | \( 1 - 39.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 91.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 37.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 46.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 120.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 27.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 81.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 10.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 205.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 115.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 312.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 90.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + 550.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 630.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 661.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 494.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 566.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 49.4T + 4.93e5T^{2} \) |
| 83 | \( 1 - 564.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 464.T + 9.12e5T^{2} \) |
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
−13.16151332510959660812793448131, −11.83923611797951373656279152319, −10.85748455458594214494715837847, −9.313960582537650658246706482761, −8.448906535760360994184158615452, −7.934139585744887966745795652895, −6.54501406195446523068143946475, −4.14131811639009938023043855460, −3.57225340736541773100852426548, −1.69347022606794249819762804062,
1.69347022606794249819762804062, 3.57225340736541773100852426548, 4.14131811639009938023043855460, 6.54501406195446523068143946475, 7.934139585744887966745795652895, 8.448906535760360994184158615452, 9.313960582537650658246706482761, 10.85748455458594214494715837847, 11.83923611797951373656279152319, 13.16151332510959660812793448131