| L(s) = 1 | − 3.90e3·2-s − 5.49e5·3-s + 6.87e6·4-s + 3.94e5·5-s + 2.14e9·6-s + 5.93e9·8-s + 2.07e11·9-s − 1.53e9·10-s + 1.32e12·11-s − 3.77e12·12-s + 3.67e12·13-s − 2.16e11·15-s − 8.07e13·16-s + 2.14e14·17-s − 8.11e14·18-s − 2.08e14·19-s + 2.70e12·20-s − 5.15e15·22-s + 3.53e15·23-s − 3.25e15·24-s − 1.19e16·25-s − 1.43e16·26-s − 6.24e16·27-s − 1.49e16·29-s + 8.45e14·30-s + 2.46e17·31-s + 2.65e17·32-s + ⋯ |
| L(s) = 1 | − 1.34·2-s − 1.79·3-s + 0.819·4-s + 0.00360·5-s + 2.41·6-s + 0.244·8-s + 2.20·9-s − 0.00486·10-s + 1.39·11-s − 1.46·12-s + 0.569·13-s − 0.00646·15-s − 1.14·16-s + 1.51·17-s − 2.97·18-s − 0.411·19-s + 0.00295·20-s − 1.88·22-s + 0.772·23-s − 0.437·24-s − 0.999·25-s − 0.767·26-s − 2.16·27-s − 0.228·29-s + 0.00871·30-s + 1.74·31-s + 1.30·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| good | 2 | \( 1 + 3.90e3T + 8.38e6T^{2} \) |
| 3 | \( 1 + 5.49e5T + 9.41e10T^{2} \) |
| 5 | \( 1 - 3.94e5T + 1.19e16T^{2} \) |
| 11 | \( 1 - 1.32e12T + 8.95e23T^{2} \) |
| 13 | \( 1 - 3.67e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 2.14e14T + 1.99e28T^{2} \) |
| 19 | \( 1 + 2.08e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 3.53e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 1.49e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 2.46e17T + 2.00e34T^{2} \) |
| 37 | \( 1 + 1.67e18T + 1.17e36T^{2} \) |
| 41 | \( 1 + 4.86e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 9.26e18T + 3.71e37T^{2} \) |
| 47 | \( 1 - 2.21e19T + 2.87e38T^{2} \) |
| 53 | \( 1 - 9.32e18T + 4.55e39T^{2} \) |
| 59 | \( 1 - 1.70e20T + 5.36e40T^{2} \) |
| 61 | \( 1 - 7.54e19T + 1.15e41T^{2} \) |
| 67 | \( 1 + 4.83e20T + 9.99e41T^{2} \) |
| 71 | \( 1 - 3.87e20T + 3.79e42T^{2} \) |
| 73 | \( 1 - 1.10e21T + 7.18e42T^{2} \) |
| 79 | \( 1 + 8.90e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 1.98e22T + 1.37e44T^{2} \) |
| 89 | \( 1 + 2.68e22T + 6.85e44T^{2} \) |
| 97 | \( 1 + 8.68e22T + 4.96e45T^{2} \) |
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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
−10.44685099818090272105097365401, −9.777760896155377994047745100506, −8.474494315453548394491692046780, −7.09694663891304192393674149427, −6.32351676381471099843263226624, −5.14211741587443484675221316252, −3.86872922098441968944919994242, −1.51097905146349226089883526377, −1.01398850398613063084472920947, 0,
1.01398850398613063084472920947, 1.51097905146349226089883526377, 3.86872922098441968944919994242, 5.14211741587443484675221316252, 6.32351676381471099843263226624, 7.09694663891304192393674149427, 8.474494315453548394491692046780, 9.777760896155377994047745100506, 10.44685099818090272105097365401
