| L(s) = 1 | − 2.79e3·2-s − 3.70e5·3-s − 5.80e5·4-s + 4.88e7·5-s + 1.03e9·6-s + 2.05e9·7-s + 2.50e10·8-s + 4.32e10·9-s − 1.36e11·10-s + 6.42e11·11-s + 2.15e11·12-s + 3.10e12·13-s − 5.74e12·14-s − 1.80e13·15-s − 6.51e13·16-s − 2.44e14·17-s − 1.20e14·18-s + 3.84e14·19-s − 2.83e13·20-s − 7.62e14·21-s − 1.79e15·22-s − 4.73e15·23-s − 9.28e15·24-s + 2.38e15·25-s − 8.67e15·26-s + 1.88e16·27-s − 1.19e15·28-s + ⋯ |
| L(s) = 1 | − 0.964·2-s − 1.20·3-s − 0.0691·4-s + 0.447·5-s + 1.16·6-s + 0.393·7-s + 1.03·8-s + 0.459·9-s − 0.431·10-s + 0.679·11-s + 0.0835·12-s + 0.480·13-s − 0.379·14-s − 0.540·15-s − 0.926·16-s − 1.72·17-s − 0.443·18-s + 0.756·19-s − 0.0309·20-s − 0.474·21-s − 0.655·22-s − 1.03·23-s − 1.24·24-s + 0.200·25-s − 0.463·26-s + 0.653·27-s − 0.0271·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 4.88e7T \) |
| good | 2 | \( 1 + 2.79e3T + 8.38e6T^{2} \) |
| 3 | \( 1 + 3.70e5T + 9.41e10T^{2} \) |
| 7 | \( 1 - 2.05e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 6.42e11T + 8.95e23T^{2} \) |
| 13 | \( 1 - 3.10e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 2.44e14T + 1.99e28T^{2} \) |
| 19 | \( 1 - 3.84e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 4.73e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 1.15e17T + 4.31e33T^{2} \) |
| 31 | \( 1 - 4.06e16T + 2.00e34T^{2} \) |
| 37 | \( 1 + 1.35e18T + 1.17e36T^{2} \) |
| 41 | \( 1 + 5.05e17T + 1.24e37T^{2} \) |
| 43 | \( 1 + 5.96e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + 2.55e19T + 2.87e38T^{2} \) |
| 53 | \( 1 - 9.91e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 2.33e20T + 5.36e40T^{2} \) |
| 61 | \( 1 + 1.54e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 5.45e20T + 9.99e41T^{2} \) |
| 71 | \( 1 + 2.35e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 7.49e20T + 7.18e42T^{2} \) |
| 79 | \( 1 + 7.69e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 8.79e21T + 1.37e44T^{2} \) |
| 89 | \( 1 - 2.96e22T + 6.85e44T^{2} \) |
| 97 | \( 1 + 6.40e22T + 4.96e45T^{2} \) |
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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
−17.55841549127832474083883499368, −16.19559586326048970169002729758, −13.74090675738545802532976829209, −11.64097412499975473174530099918, −10.31434744629029332781650606138, −8.662072177564089487383224736030, −6.51536500727386182175920052570, −4.73692437868497777512787410532, −1.43037514122873215593986529295, 0,
1.43037514122873215593986529295, 4.73692437868497777512787410532, 6.51536500727386182175920052570, 8.662072177564089487383224736030, 10.31434744629029332781650606138, 11.64097412499975473174530099918, 13.74090675738545802532976829209, 16.19559586326048970169002729758, 17.55841549127832474083883499368
