| L(s) = 1 | − 1.22·2-s − 3·3-s − 6.49·4-s − 2.10·5-s + 3.67·6-s + 35.0·7-s + 17.7·8-s + 9·9-s + 2.57·10-s − 11.3·11-s + 19.4·12-s − 54.5·13-s − 42.9·14-s + 6.30·15-s + 30.1·16-s − 9.39·17-s − 11.0·18-s − 15.5·19-s + 13.6·20-s − 105.·21-s + 13.9·22-s + 183.·23-s − 53.3·24-s − 120.·25-s + 66.8·26-s − 27·27-s − 227.·28-s + ⋯ |
| L(s) = 1 | − 0.433·2-s − 0.577·3-s − 0.812·4-s − 0.187·5-s + 0.250·6-s + 1.89·7-s + 0.785·8-s + 0.333·9-s + 0.0814·10-s − 0.312·11-s + 0.468·12-s − 1.16·13-s − 0.819·14-s + 0.108·15-s + 0.471·16-s − 0.134·17-s − 0.144·18-s − 0.187·19-s + 0.152·20-s − 1.09·21-s + 0.135·22-s + 1.66·23-s − 0.453·24-s − 0.964·25-s + 0.504·26-s − 0.192·27-s − 1.53·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.038432544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.038432544\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 157 | \( 1 + 157T \) |
| good | 2 | \( 1 + 1.22T + 8T^{2} \) |
| 5 | \( 1 + 2.10T + 125T^{2} \) |
| 7 | \( 1 - 35.0T + 343T^{2} \) |
| 11 | \( 1 + 11.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 54.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 9.39T + 4.91e3T^{2} \) |
| 19 | \( 1 + 15.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 183.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 139.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 236.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 156.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 182.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 423.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 439.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 497.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 216.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 325.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 505.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 993.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 370.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 810.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 654.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.07e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 844.T + 9.12e5T^{2} \) |
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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
−10.92334922925860256926247771240, −9.611579462325000646901959013657, −8.866122213568904118018444219490, −7.70636680183074571974347030642, −7.44449165727997523598889921347, −5.53635121380014340155480682142, −4.92459036433605282495645162487, −4.12381918264327531471975458150, −2.04474083632926885350660175701, −0.72607204872733115280191801975,
0.72607204872733115280191801975, 2.04474083632926885350660175701, 4.12381918264327531471975458150, 4.92459036433605282495645162487, 5.53635121380014340155480682142, 7.44449165727997523598889921347, 7.70636680183074571974347030642, 8.866122213568904118018444219490, 9.611579462325000646901959013657, 10.92334922925860256926247771240
