| L(s) = 1 | − 3-s − 3.70·5-s + 7-s + 9-s + 5.70·13-s + 3.70·15-s + 5.40·19-s − 21-s + 23-s + 8.70·25-s − 27-s + 0.298·29-s − 6·31-s − 3.70·35-s + 3.70·37-s − 5.70·39-s + 7.70·41-s − 5.70·43-s − 3.70·45-s − 3.70·47-s + 49-s + 9.40·53-s − 5.40·57-s + 0.596·59-s + 10·61-s + 63-s − 21.1·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.65·5-s + 0.377·7-s + 0.333·9-s + 1.58·13-s + 0.955·15-s + 1.23·19-s − 0.218·21-s + 0.208·23-s + 1.74·25-s − 0.192·27-s + 0.0554·29-s − 1.07·31-s − 0.625·35-s + 0.608·37-s − 0.912·39-s + 1.20·41-s − 0.869·43-s − 0.551·45-s − 0.539·47-s + 0.142·49-s + 1.29·53-s − 0.715·57-s + 0.0777·59-s + 1.28·61-s + 0.125·63-s − 2.61·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.276188597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.276188597\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 3.70T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 5.70T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 5.40T + 19T^{2} \) |
| 29 | \( 1 - 0.298T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 3.70T + 37T^{2} \) |
| 41 | \( 1 - 7.70T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 + 3.70T + 47T^{2} \) |
| 53 | \( 1 - 9.40T + 53T^{2} \) |
| 59 | \( 1 - 0.596T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 7.40T + 71T^{2} \) |
| 73 | \( 1 + 5.40T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 2.59T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + 1.10T + 97T^{2} \) |
| 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
−7.70742258078336189751581789613, −7.34675305516674728465722927261, −6.54502222960241848762298354454, −5.70865383578022814440781750089, −5.05763251533282894078312143265, −4.14058995600761361184270024139, −3.75461835833087371428056260143, −2.93397756191829850477995034200, −1.44027136286692970818594994674, −0.63908888557670333050335651406,
0.63908888557670333050335651406, 1.44027136286692970818594994674, 2.93397756191829850477995034200, 3.75461835833087371428056260143, 4.14058995600761361184270024139, 5.05763251533282894078312143265, 5.70865383578022814440781750089, 6.54502222960241848762298354454, 7.34675305516674728465722927261, 7.70742258078336189751581789613
