L(s) = 1 | − 5.11e5·3-s + 6.64e7·5-s + 8.09e9·7-s + 1.67e11·9-s + 1.78e11·11-s + 1.22e13·13-s − 3.39e13·15-s − 1.77e14·17-s + 7.96e14·19-s − 4.14e15·21-s − 2.89e15·23-s − 7.50e15·25-s − 3.77e16·27-s − 4.86e16·29-s − 1.32e17·31-s − 9.12e16·33-s + 5.37e17·35-s − 5.80e17·37-s − 6.27e18·39-s − 2.82e18·41-s + 1.78e18·43-s + 1.11e19·45-s − 1.39e19·47-s + 3.82e19·49-s + 9.06e19·51-s − 1.07e20·53-s + 1.18e19·55-s + ⋯ |
L(s) = 1 | − 1.66·3-s + 0.608·5-s + 1.54·7-s + 1.78·9-s + 0.188·11-s + 1.89·13-s − 1.01·15-s − 1.25·17-s + 1.56·19-s − 2.58·21-s − 0.632·23-s − 0.629·25-s − 1.30·27-s − 0.739·29-s − 0.937·31-s − 0.314·33-s + 0.941·35-s − 0.536·37-s − 3.16·39-s − 0.802·41-s + 0.292·43-s + 1.08·45-s − 0.820·47-s + 1.39·49-s + 2.09·51-s − 1.59·53-s + 0.114·55-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
\(L(12)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{25}{2})\) |
|
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 | \( 1 + 5.11e5T + 9.41e10T^{2} \) |
| 5 | \( 1 - 6.64e7T + 1.19e16T^{2} \) |
| 7 | \( 1 - 8.09e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 1.78e11T + 8.95e23T^{2} \) |
| 13 | \( 1 - 1.22e13T + 4.17e25T^{2} \) |
| 17 | \( 1 + 1.77e14T + 1.99e28T^{2} \) |
| 19 | \( 1 - 7.96e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 2.89e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 4.86e16T + 4.31e33T^{2} \) |
| 31 | \( 1 + 1.32e17T + 2.00e34T^{2} \) |
| 37 | \( 1 + 5.80e17T + 1.17e36T^{2} \) |
| 41 | \( 1 + 2.82e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 1.78e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + 1.39e19T + 2.87e38T^{2} \) |
| 53 | \( 1 + 1.07e20T + 4.55e39T^{2} \) |
| 59 | \( 1 - 5.27e19T + 5.36e40T^{2} \) |
| 61 | \( 1 + 1.12e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 1.35e21T + 9.99e41T^{2} \) |
| 71 | \( 1 - 4.67e20T + 3.79e42T^{2} \) |
| 73 | \( 1 - 2.11e20T + 7.18e42T^{2} \) |
| 79 | \( 1 - 1.15e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 1.97e22T + 1.37e44T^{2} \) |
| 89 | \( 1 + 4.49e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 7.68e22T + 4.96e45T^{2} \) |
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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
−10.57572472126345712988224709377, −9.159745765896552521164332725584, −7.84376558425581498562695535466, −6.52767489174579572205820927696, −5.68857251563522981631332257453, −4.97608933320919175985850488180, −3.85760554764912384823317051291, −1.70233288118952416423590996189, −1.31187778212415735071759100028, 0,
1.31187778212415735071759100028, 1.70233288118952416423590996189, 3.85760554764912384823317051291, 4.97608933320919175985850488180, 5.68857251563522981631332257453, 6.52767489174579572205820927696, 7.84376558425581498562695535466, 9.159745765896552521164332725584, 10.57572472126345712988224709377