| L(s) = 1 | − 3.49·2-s − 4.75·3-s + 4.24·4-s + 6.95·5-s + 16.6·6-s + 9.95·7-s + 13.1·8-s − 4.34·9-s − 24.3·10-s − 3.72·11-s − 20.1·12-s + 64.1·13-s − 34.8·14-s − 33.1·15-s − 79.9·16-s + 111.·17-s + 15.2·18-s + 134.·19-s + 29.5·20-s − 47.3·21-s + 13.0·22-s − 62.5·24-s − 76.5·25-s − 224.·26-s + 149.·27-s + 42.2·28-s − 74.2·29-s + ⋯ |
| L(s) = 1 | − 1.23·2-s − 0.915·3-s + 0.530·4-s + 0.622·5-s + 1.13·6-s + 0.537·7-s + 0.581·8-s − 0.160·9-s − 0.769·10-s − 0.101·11-s − 0.485·12-s + 1.36·13-s − 0.664·14-s − 0.570·15-s − 1.24·16-s + 1.58·17-s + 0.199·18-s + 1.62·19-s + 0.330·20-s − 0.492·21-s + 0.126·22-s − 0.532·24-s − 0.612·25-s − 1.69·26-s + 1.06·27-s + 0.284·28-s − 0.475·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8785518635\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8785518635\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 + 3.49T + 8T^{2} \) |
| 3 | \( 1 + 4.75T + 27T^{2} \) |
| 5 | \( 1 - 6.95T + 125T^{2} \) |
| 7 | \( 1 - 9.95T + 343T^{2} \) |
| 11 | \( 1 + 3.72T + 1.33e3T^{2} \) |
| 13 | \( 1 - 64.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 111.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 134.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 74.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 205.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 155.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 252.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 252.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 435.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 383.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 584.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 746.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 298.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 458.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 293.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 804.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 219.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 972.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 657.T + 9.12e5T^{2} \) |
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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
−10.25259456940613492347253236312, −9.710178531838387365710986431233, −8.685267734727491012375985692615, −7.937821924116143031944711058808, −6.96698693643008865740818084496, −5.68555935010689831890187966307, −5.26325541235090650025820103810, −3.54559415060557543086600060758, −1.66683070944278179918882715670, −0.78558237608970070348816471180,
0.78558237608970070348816471180, 1.66683070944278179918882715670, 3.54559415060557543086600060758, 5.26325541235090650025820103810, 5.68555935010689831890187966307, 6.96698693643008865740818084496, 7.937821924116143031944711058808, 8.685267734727491012375985692615, 9.710178531838387365710986431233, 10.25259456940613492347253236312
