| L(s) = 1 | + 3-s + 2·5-s + 7-s − 2·9-s + 11-s + 13-s + 2·15-s + 8·17-s − 6·19-s + 21-s − 5·23-s − 25-s − 5·27-s − 29-s + 31-s + 33-s + 2·35-s − 8·37-s + 39-s − 8·41-s − 43-s − 4·45-s + 49-s + 8·51-s + 11·53-s + 2·55-s − 6·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.377·7-s − 2/3·9-s + 0.301·11-s + 0.277·13-s + 0.516·15-s + 1.94·17-s − 1.37·19-s + 0.218·21-s − 1.04·23-s − 1/5·25-s − 0.962·27-s − 0.185·29-s + 0.179·31-s + 0.174·33-s + 0.338·35-s − 1.31·37-s + 0.160·39-s − 1.24·41-s − 0.152·43-s − 0.596·45-s + 1/7·49-s + 1.12·51-s + 1.51·53-s + 0.269·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.164449543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.164449543\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
| 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
−13.52883257706535, −13.12126621687371, −12.23140056837001, −12.09610326634595, −11.60323991855070, −10.88244947469044, −10.40515115263163, −9.908125344082103, −9.727606056587872, −8.805589075796430, −8.579083854469243, −8.260579733506374, −7.493540730959074, −7.161326139077326, −6.237939796832465, −5.956878665142090, −5.549160099890345, −4.963329408905921, −4.207459439759406, −3.601080634597858, −3.212665230692469, −2.432226646785312, −1.865999925700556, −1.512646983329125, −0.4779403771926334,
0.4779403771926334, 1.512646983329125, 1.865999925700556, 2.432226646785312, 3.212665230692469, 3.601080634597858, 4.207459439759406, 4.963329408905921, 5.549160099890345, 5.956878665142090, 6.237939796832465, 7.161326139077326, 7.493540730959074, 8.260579733506374, 8.579083854469243, 8.805589075796430, 9.727606056587872, 9.908125344082103, 10.40515115263163, 10.88244947469044, 11.60323991855070, 12.09610326634595, 12.23140056837001, 13.12126621687371, 13.52883257706535
