| L(s) = 1 | − 2.56·3-s − 5-s + 1.56·7-s + 3.56·9-s − 2·11-s + 0.561·13-s + 2.56·15-s + 5.56·17-s − 2·19-s − 4·21-s − 23-s + 25-s − 1.43·27-s + 0.123·29-s − 8.12·31-s + 5.12·33-s − 1.56·35-s − 3.56·37-s − 1.43·39-s − 4.12·41-s − 10.2·43-s − 3.56·45-s + 3.68·47-s − 4.56·49-s − 14.2·51-s + 4.43·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 1.47·3-s − 0.447·5-s + 0.590·7-s + 1.18·9-s − 0.603·11-s + 0.155·13-s + 0.661·15-s + 1.34·17-s − 0.458·19-s − 0.872·21-s − 0.208·23-s + 0.200·25-s − 0.276·27-s + 0.0228·29-s − 1.45·31-s + 0.891·33-s − 0.263·35-s − 0.585·37-s − 0.230·39-s − 0.643·41-s − 1.56·43-s − 0.530·45-s + 0.537·47-s − 0.651·49-s − 1.99·51-s + 0.609·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 0.561T + 13T^{2} \) |
| 17 | \( 1 - 5.56T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 - 0.123T + 29T^{2} \) |
| 31 | \( 1 + 8.12T + 31T^{2} \) |
| 37 | \( 1 + 3.56T + 37T^{2} \) |
| 41 | \( 1 + 4.12T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 3.68T + 47T^{2} \) |
| 53 | \( 1 - 4.43T + 53T^{2} \) |
| 59 | \( 1 + 5.56T + 59T^{2} \) |
| 61 | \( 1 + 9.12T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 + 5T + 71T^{2} \) |
| 73 | \( 1 - 3.43T + 73T^{2} \) |
| 79 | \( 1 + 9.12T + 79T^{2} \) |
| 83 | \( 1 + 4.68T + 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 + 3.12T + 97T^{2} \) |
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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
−10.04235762469178362002793081819, −8.744167355256539290499300744553, −7.82217264057467387909205067532, −7.08046810530948295305554100968, −6.00307995101330595635695902458, −5.32244973383612532078712401436, −4.58372953865440310153755007996, −3.34803918581233743638264399853, −1.54170528812686804589089880290, 0,
1.54170528812686804589089880290, 3.34803918581233743638264399853, 4.58372953865440310153755007996, 5.32244973383612532078712401436, 6.00307995101330595635695902458, 7.08046810530948295305554100968, 7.82217264057467387909205067532, 8.744167355256539290499300744553, 10.04235762469178362002793081819
