| L(s) = 1 | − 5.74e5·3-s + 8.41e7·5-s − 5.95e9·7-s + 2.35e11·9-s + 1.24e12·11-s + 7.21e12·13-s − 4.83e13·15-s − 6.50e13·17-s − 5.56e14·19-s + 3.42e15·21-s − 3.08e15·23-s − 4.84e15·25-s − 8.12e16·27-s − 4.69e16·29-s + 2.40e17·31-s − 7.13e17·33-s − 5.01e17·35-s − 5.28e17·37-s − 4.14e18·39-s + 1.59e18·41-s + 7.35e18·43-s + 1.98e19·45-s − 1.63e19·47-s + 8.12e18·49-s + 3.73e19·51-s − 2.64e19·53-s + 1.04e20·55-s + ⋯ |
| L(s) = 1 | − 1.87·3-s + 0.770·5-s − 1.13·7-s + 2.50·9-s + 1.31·11-s + 1.11·13-s − 1.44·15-s − 0.460·17-s − 1.09·19-s + 2.13·21-s − 0.674·23-s − 0.406·25-s − 2.81·27-s − 0.715·29-s + 1.70·31-s − 2.45·33-s − 0.877·35-s − 0.488·37-s − 2.08·39-s + 0.453·41-s + 1.20·43-s + 1.92·45-s − 0.967·47-s + 0.296·49-s + 0.862·51-s − 0.391·53-s + 1.01·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 5.74e5T + 9.41e10T^{2} \) |
| 5 | \( 1 - 8.41e7T + 1.19e16T^{2} \) |
| 7 | \( 1 + 5.95e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 1.24e12T + 8.95e23T^{2} \) |
| 13 | \( 1 - 7.21e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 6.50e13T + 1.99e28T^{2} \) |
| 19 | \( 1 + 5.56e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 3.08e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 4.69e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 2.40e17T + 2.00e34T^{2} \) |
| 37 | \( 1 + 5.28e17T + 1.17e36T^{2} \) |
| 41 | \( 1 - 1.59e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 7.35e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + 1.63e19T + 2.87e38T^{2} \) |
| 53 | \( 1 + 2.64e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 3.24e20T + 5.36e40T^{2} \) |
| 61 | \( 1 + 3.22e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 1.09e21T + 9.99e41T^{2} \) |
| 71 | \( 1 + 2.30e20T + 3.79e42T^{2} \) |
| 73 | \( 1 - 8.33e20T + 7.18e42T^{2} \) |
| 79 | \( 1 - 2.59e21T + 4.42e43T^{2} \) |
| 83 | \( 1 + 1.05e22T + 1.37e44T^{2} \) |
| 89 | \( 1 - 4.07e22T + 6.85e44T^{2} \) |
| 97 | \( 1 + 9.40e22T + 4.96e45T^{2} \) |
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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
−15.80962119748965286284403023803, −13.39229671775951288574836396720, −12.14946864444288184488189567580, −10.79524661974056312355186155720, −9.521333393595047162042737082394, −6.42466379102561216966283980141, −6.10621489963350830003614920012, −4.15551091387888452174319835526, −1.42610746650890802960986163753, 0,
1.42610746650890802960986163753, 4.15551091387888452174319835526, 6.10621489963350830003614920012, 6.42466379102561216966283980141, 9.521333393595047162042737082394, 10.79524661974056312355186155720, 12.14946864444288184488189567580, 13.39229671775951288574836396720, 15.80962119748965286284403023803
