| L(s) = 1 | − 2.64·2-s + 5.00·4-s − 2.64·5-s − 7.93·8-s + 7.00·10-s + 2.64·11-s + 2·13-s + 11.0·16-s − 7·19-s − 13.2·20-s − 7.00·22-s + 7.93·23-s + 2.00·25-s − 5.29·26-s − 5.29·29-s − 3·31-s − 13.2·32-s − 3·37-s + 18.5·38-s + 21.0·40-s + 2.64·41-s + 8·43-s + 13.2·44-s − 21.0·46-s − 5.29·50-s + 10.0·52-s − 7.00·55-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 2.50·4-s − 1.18·5-s − 2.80·8-s + 2.21·10-s + 0.797·11-s + 0.554·13-s + 2.75·16-s − 1.60·19-s − 2.95·20-s − 1.49·22-s + 1.65·23-s + 0.400·25-s − 1.03·26-s − 0.982·29-s − 0.538·31-s − 2.33·32-s − 0.493·37-s + 3.00·38-s + 3.32·40-s + 0.413·41-s + 1.21·43-s + 1.99·44-s − 3.09·46-s − 0.748·50-s + 1.38·52-s − 0.943·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4808989353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4808989353\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.64T + 2T^{2} \) |
| 5 | \( 1 + 2.64T + 5T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 - 7.93T + 23T^{2} \) |
| 29 | \( 1 + 5.29T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 - 2.64T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 + 7.93T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + 18.5T + 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.302885658540947908315063634495, −8.848276503002121803923774537655, −8.214090289079762875375570043870, −7.35744070570776281569178494331, −6.84416069498005048775938673381, −5.89795357588904636549185706215, −4.26299173805444489815711329126, −3.27482901584446912986752730007, −1.93156615238358713005091899982, −0.66453989430569328623377113240,
0.66453989430569328623377113240, 1.93156615238358713005091899982, 3.27482901584446912986752730007, 4.26299173805444489815711329126, 5.89795357588904636549185706215, 6.84416069498005048775938673381, 7.35744070570776281569178494331, 8.214090289079762875375570043870, 8.848276503002121803923774537655, 9.302885658540947908315063634495
