| L(s) = 1 | − 5.99e5·3-s − 1.94e8·5-s − 4.67e9·7-s + 2.65e11·9-s − 9.07e11·11-s + 6.04e12·13-s + 1.16e14·15-s − 2.80e13·17-s + 9.06e13·19-s + 2.80e15·21-s − 2.89e14·23-s + 2.57e16·25-s − 1.02e17·27-s + 4.09e16·29-s + 1.20e17·31-s + 5.44e17·33-s + 9.07e17·35-s − 1.40e18·37-s − 3.62e18·39-s + 5.50e17·41-s − 8.04e18·43-s − 5.15e19·45-s − 2.67e18·47-s − 5.50e18·49-s + 1.68e19·51-s − 4.15e19·53-s + 1.76e20·55-s + ⋯ |
| L(s) = 1 | − 1.95·3-s − 1.77·5-s − 0.893·7-s + 2.82·9-s − 0.959·11-s + 0.935·13-s + 3.47·15-s − 0.198·17-s + 0.178·19-s + 1.74·21-s − 0.0633·23-s + 2.16·25-s − 3.56·27-s + 0.622·29-s + 0.853·31-s + 1.87·33-s + 1.58·35-s − 1.29·37-s − 1.82·39-s + 0.156·41-s − 1.32·43-s − 5.01·45-s − 0.157·47-s − 0.201·49-s + 0.387·51-s − 0.615·53-s + 1.70·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(0.07081536984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07081536984\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 5.99e5T + 9.41e10T^{2} \) |
| 5 | \( 1 + 1.94e8T + 1.19e16T^{2} \) |
| 7 | \( 1 + 4.67e9T + 2.73e19T^{2} \) |
| 11 | \( 1 + 9.07e11T + 8.95e23T^{2} \) |
| 13 | \( 1 - 6.04e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 2.80e13T + 1.99e28T^{2} \) |
| 19 | \( 1 - 9.06e13T + 2.57e29T^{2} \) |
| 23 | \( 1 + 2.89e14T + 2.08e31T^{2} \) |
| 29 | \( 1 - 4.09e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 1.20e17T + 2.00e34T^{2} \) |
| 37 | \( 1 + 1.40e18T + 1.17e36T^{2} \) |
| 41 | \( 1 - 5.50e17T + 1.24e37T^{2} \) |
| 43 | \( 1 + 8.04e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + 2.67e18T + 2.87e38T^{2} \) |
| 53 | \( 1 + 4.15e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 1.95e20T + 5.36e40T^{2} \) |
| 61 | \( 1 - 3.54e20T + 1.15e41T^{2} \) |
| 67 | \( 1 - 1.08e21T + 9.99e41T^{2} \) |
| 71 | \( 1 - 2.49e20T + 3.79e42T^{2} \) |
| 73 | \( 1 + 4.40e21T + 7.18e42T^{2} \) |
| 79 | \( 1 + 9.78e21T + 4.42e43T^{2} \) |
| 83 | \( 1 + 2.67e20T + 1.37e44T^{2} \) |
| 89 | \( 1 - 1.67e22T + 6.85e44T^{2} \) |
| 97 | \( 1 + 9.64e22T + 4.96e45T^{2} \) |
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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
−10.91042649929025723753734918818, −10.07672569105980793539640205199, −8.267079531916067876541166806038, −7.12728575879301569843871317911, −6.37911554333460882375170433085, −5.18140944095898623997481471239, −4.27004584012120820119798293237, −3.28273731238815637526021124821, −1.16083673826385432179516581958, −0.14465473799707158873997382577,
0.14465473799707158873997382577, 1.16083673826385432179516581958, 3.28273731238815637526021124821, 4.27004584012120820119798293237, 5.18140944095898623997481471239, 6.37911554333460882375170433085, 7.12728575879301569843871317911, 8.267079531916067876541166806038, 10.07672569105980793539640205199, 10.91042649929025723753734918818
