| L(s) = 1 | − 4.01e3·2-s − 3.88e5·3-s + 7.74e6·4-s − 1.05e8·5-s + 1.56e9·6-s + 2.59e9·8-s + 5.67e10·9-s + 4.21e11·10-s + 2.52e11·11-s − 3.00e12·12-s + 3.59e12·13-s + 4.08e13·15-s − 7.53e13·16-s − 2.34e14·17-s − 2.27e14·18-s + 6.23e14·19-s − 8.13e14·20-s − 1.01e15·22-s − 3.58e15·23-s − 1.00e15·24-s − 8.82e14·25-s − 1.44e16·26-s + 1.45e16·27-s − 2.05e16·29-s − 1.63e17·30-s − 1.36e17·31-s + 2.80e17·32-s + ⋯ |
| L(s) = 1 | − 1.38·2-s − 1.26·3-s + 0.922·4-s − 0.962·5-s + 1.75·6-s + 0.106·8-s + 0.602·9-s + 1.33·10-s + 0.266·11-s − 1.16·12-s + 0.555·13-s + 1.21·15-s − 1.07·16-s − 1.65·17-s − 0.835·18-s + 1.22·19-s − 0.888·20-s − 0.369·22-s − 0.785·23-s − 0.135·24-s − 0.0740·25-s − 0.770·26-s + 0.502·27-s − 0.313·29-s − 1.68·30-s − 0.963·31-s + 1.37·32-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 | \( 1 + 4.01e3T + 8.38e6T^{2} \) |
| 3 | \( 1 + 3.88e5T + 9.41e10T^{2} \) |
| 5 | \( 1 + 1.05e8T + 1.19e16T^{2} \) |
| 11 | \( 1 - 2.52e11T + 8.95e23T^{2} \) |
| 13 | \( 1 - 3.59e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 2.34e14T + 1.99e28T^{2} \) |
| 19 | \( 1 - 6.23e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 3.58e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 2.05e16T + 4.31e33T^{2} \) |
| 31 | \( 1 + 1.36e17T + 2.00e34T^{2} \) |
| 37 | \( 1 + 1.23e18T + 1.17e36T^{2} \) |
| 41 | \( 1 + 1.40e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 2.18e17T + 3.71e37T^{2} \) |
| 47 | \( 1 - 8.67e18T + 2.87e38T^{2} \) |
| 53 | \( 1 + 7.63e19T + 4.55e39T^{2} \) |
| 59 | \( 1 - 1.01e18T + 5.36e40T^{2} \) |
| 61 | \( 1 + 2.87e20T + 1.15e41T^{2} \) |
| 67 | \( 1 - 1.47e21T + 9.99e41T^{2} \) |
| 71 | \( 1 - 7.64e20T + 3.79e42T^{2} \) |
| 73 | \( 1 - 3.49e21T + 7.18e42T^{2} \) |
| 79 | \( 1 - 1.02e22T + 4.42e43T^{2} \) |
| 83 | \( 1 - 7.71e21T + 1.37e44T^{2} \) |
| 89 | \( 1 + 4.58e21T + 6.85e44T^{2} \) |
| 97 | \( 1 - 1.13e23T + 4.96e45T^{2} \) |
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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
−10.81591486587553546815721792259, −9.468198422406310202469890816024, −8.414063038711063383699037278229, −7.34432700308319776646433506223, −6.38792853132450593755718084761, −4.98420642922831133984871789837, −3.77607373681917342487380261656, −1.85629561139771802680929282611, −0.67599019628767350974839327353, 0,
0.67599019628767350974839327353, 1.85629561139771802680929282611, 3.77607373681917342487380261656, 4.98420642922831133984871789837, 6.38792853132450593755718084761, 7.34432700308319776646433506223, 8.414063038711063383699037278229, 9.468198422406310202469890816024, 10.81591486587553546815721792259
