| L(s) = 1 | + 9·3-s − 24·5-s − 68·7-s + 81·9-s − 452·11-s − 1.05e3·13-s − 216·15-s − 418·17-s − 2.49e3·19-s − 612·21-s + 2.39e3·23-s − 2.54e3·25-s + 729·27-s + 1.52e3·29-s + 1.94e3·31-s − 4.06e3·33-s + 1.63e3·35-s + 8.97e3·37-s − 9.46e3·39-s + 1.51e4·41-s + 2.68e3·43-s − 1.94e3·45-s + 1.07e4·47-s − 1.21e4·49-s − 3.76e3·51-s + 8.04e3·53-s + 1.08e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.429·5-s − 0.524·7-s + 1/3·9-s − 1.12·11-s − 1.72·13-s − 0.247·15-s − 0.350·17-s − 1.58·19-s − 0.302·21-s + 0.942·23-s − 0.815·25-s + 0.192·27-s + 0.335·29-s + 0.364·31-s − 0.650·33-s + 0.225·35-s + 1.07·37-s − 0.996·39-s + 1.40·41-s + 0.221·43-s − 0.143·45-s + 0.709·47-s − 0.724·49-s − 0.202·51-s + 0.393·53-s + 0.483·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.128274548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.128274548\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 24 T + p^{5} T^{2} \) |
| 7 | \( 1 + 68 T + p^{5} T^{2} \) |
| 11 | \( 1 + 452 T + p^{5} T^{2} \) |
| 13 | \( 1 + 1052 T + p^{5} T^{2} \) |
| 17 | \( 1 + 418 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2492 T + p^{5} T^{2} \) |
| 23 | \( 1 - 104 p T + p^{5} T^{2} \) |
| 29 | \( 1 - 1520 T + p^{5} T^{2} \) |
| 31 | \( 1 - 1948 T + p^{5} T^{2} \) |
| 37 | \( 1 - 8972 T + p^{5} T^{2} \) |
| 41 | \( 1 - 15174 T + p^{5} T^{2} \) |
| 43 | \( 1 - 2684 T + p^{5} T^{2} \) |
| 47 | \( 1 - 10744 T + p^{5} T^{2} \) |
| 53 | \( 1 - 8048 T + p^{5} T^{2} \) |
| 59 | \( 1 - 13356 T + p^{5} T^{2} \) |
| 61 | \( 1 - 19260 T + p^{5} T^{2} \) |
| 67 | \( 1 + 36588 T + p^{5} T^{2} \) |
| 71 | \( 1 + 63832 T + p^{5} T^{2} \) |
| 73 | \( 1 - 14106 T + p^{5} T^{2} \) |
| 79 | \( 1 - 1252 p T + p^{5} T^{2} \) |
| 83 | \( 1 + 63292 T + p^{5} T^{2} \) |
| 89 | \( 1 + 7014 T + p^{5} T^{2} \) |
| 97 | \( 1 - 80830 T + p^{5} T^{2} \) |
| 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
−9.591259973830930091558150948057, −8.684387633543996091431988701138, −7.78179710606537025985892223849, −7.21084573991210805126700574500, −6.12911548952477338699573881691, −4.89448129274964699363691260490, −4.14423131842052485235153152160, −2.80323527829654287579124982758, −2.28047205145442454059028873694, −0.44262929028757677274015123777,
0.44262929028757677274015123777, 2.28047205145442454059028873694, 2.80323527829654287579124982758, 4.14423131842052485235153152160, 4.89448129274964699363691260490, 6.12911548952477338699573881691, 7.21084573991210805126700574500, 7.78179710606537025985892223849, 8.684387633543996091431988701138, 9.591259973830930091558150948057
