| L(s) = 1 | + 2.30·2-s − 1.47·3-s + 3.30·4-s + 0.847·5-s − 3.39·6-s + 3.00·8-s − 0.829·9-s + 1.95·10-s − 1.50·11-s − 4.86·12-s − 1.24·15-s + 0.313·16-s + 2.07·17-s − 1.90·18-s − 0.0474·19-s + 2.80·20-s − 3.46·22-s − 7.81·23-s − 4.42·24-s − 4.28·25-s + 5.64·27-s + 1.35·29-s − 2.87·30-s + 7.86·31-s − 5.29·32-s + 2.21·33-s + 4.77·34-s + ⋯ |
| L(s) = 1 | + 1.62·2-s − 0.850·3-s + 1.65·4-s + 0.378·5-s − 1.38·6-s + 1.06·8-s − 0.276·9-s + 0.617·10-s − 0.453·11-s − 1.40·12-s − 0.322·15-s + 0.0782·16-s + 0.502·17-s − 0.450·18-s − 0.0108·19-s + 0.626·20-s − 0.738·22-s − 1.63·23-s − 0.904·24-s − 0.856·25-s + 1.08·27-s + 0.252·29-s − 0.524·30-s + 1.41·31-s − 0.935·32-s + 0.385·33-s + 0.818·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.730391881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.730391881\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 3 | \( 1 + 1.47T + 3T^{2} \) |
| 5 | \( 1 - 0.847T + 5T^{2} \) |
| 11 | \( 1 + 1.50T + 11T^{2} \) |
| 17 | \( 1 - 2.07T + 17T^{2} \) |
| 19 | \( 1 + 0.0474T + 19T^{2} \) |
| 23 | \( 1 + 7.81T + 23T^{2} \) |
| 29 | \( 1 - 1.35T + 29T^{2} \) |
| 31 | \( 1 - 7.86T + 31T^{2} \) |
| 37 | \( 1 - 6.70T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 9.26T + 43T^{2} \) |
| 47 | \( 1 - 0.360T + 47T^{2} \) |
| 53 | \( 1 - 2.71T + 53T^{2} \) |
| 59 | \( 1 - 1.64T + 59T^{2} \) |
| 61 | \( 1 - 4.52T + 61T^{2} \) |
| 67 | \( 1 + 2.04T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 - 6.76T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 - 0.426T + 97T^{2} \) |
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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
−7.66819664457469855618551051775, −6.60415326542916891772315072866, −6.09083744146986008997318656129, −5.73623054117521124566251765064, −5.11752291666071020331518941224, −4.35815769363415312898334611851, −3.77442309083307995693511457488, −2.69681723534986551815302163906, −2.23237893311081306028734741547, −0.74853910896985506728218575008,
0.74853910896985506728218575008, 2.23237893311081306028734741547, 2.69681723534986551815302163906, 3.77442309083307995693511457488, 4.35815769363415312898334611851, 5.11752291666071020331518941224, 5.73623054117521124566251765064, 6.09083744146986008997318656129, 6.60415326542916891772315072866, 7.66819664457469855618551051775
