| L(s) = 1 | + 1.28e3·2-s − 5.36e5·3-s − 6.73e6·4-s − 5.01e7·5-s − 6.91e8·6-s − 1.94e10·8-s + 1.94e11·9-s − 6.45e10·10-s + 8.52e11·11-s + 3.61e12·12-s + 7.71e12·13-s + 2.69e13·15-s + 3.14e13·16-s − 1.44e13·17-s + 2.49e14·18-s + 6.89e14·19-s + 3.37e14·20-s + 1.09e15·22-s − 7.38e15·23-s + 1.04e16·24-s − 9.40e15·25-s + 9.92e15·26-s − 5.36e16·27-s − 6.36e16·29-s + 3.46e16·30-s + 4.67e14·31-s + 2.03e17·32-s + ⋯ |
| L(s) = 1 | + 0.444·2-s − 1.74·3-s − 0.802·4-s − 0.459·5-s − 0.777·6-s − 0.801·8-s + 2.06·9-s − 0.204·10-s + 0.900·11-s + 1.40·12-s + 1.19·13-s + 0.803·15-s + 0.446·16-s − 0.102·17-s + 0.916·18-s + 1.35·19-s + 0.368·20-s + 0.400·22-s − 1.61·23-s + 1.40·24-s − 0.788·25-s + 0.530·26-s − 1.85·27-s − 0.968·29-s + 0.357·30-s + 0.00330·31-s + 0.999·32-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
| \(L(12)\) |
\(\approx\) |
\(0.8327328355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8327328355\) |
| \(L(\frac{25}{2})\) |
|
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 | \( 1 - 1.28e3T + 8.38e6T^{2} \) |
| 3 | \( 1 + 5.36e5T + 9.41e10T^{2} \) |
| 5 | \( 1 + 5.01e7T + 1.19e16T^{2} \) |
| 11 | \( 1 - 8.52e11T + 8.95e23T^{2} \) |
| 13 | \( 1 - 7.71e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 1.44e13T + 1.99e28T^{2} \) |
| 19 | \( 1 - 6.89e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 7.38e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 6.36e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 4.67e14T + 2.00e34T^{2} \) |
| 37 | \( 1 + 1.83e17T + 1.17e36T^{2} \) |
| 41 | \( 1 - 3.08e17T + 1.24e37T^{2} \) |
| 43 | \( 1 - 9.97e18T + 3.71e37T^{2} \) |
| 47 | \( 1 - 8.42e18T + 2.87e38T^{2} \) |
| 53 | \( 1 - 4.40e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 1.35e20T + 5.36e40T^{2} \) |
| 61 | \( 1 - 4.44e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 8.78e20T + 9.99e41T^{2} \) |
| 71 | \( 1 + 3.16e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 4.50e21T + 7.18e42T^{2} \) |
| 79 | \( 1 + 5.63e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 7.31e21T + 1.37e44T^{2} \) |
| 89 | \( 1 - 3.89e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 1.14e22T + 4.96e45T^{2} \) |
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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
−11.57582272677838627292886785595, −10.26108211508858744484875648665, −9.098988733596959751106493271884, −7.54215703796055617334322595279, −6.09146849453590987996951925504, −5.62644415725274677589274565940, −4.30588523615670009584833614987, −3.72533684450277281430331937316, −1.36046111969763551589816518429, −0.46312538445313582743800498789,
0.46312538445313582743800498789, 1.36046111969763551589816518429, 3.72533684450277281430331937316, 4.30588523615670009584833614987, 5.62644415725274677589274565940, 6.09146849453590987996951925504, 7.54215703796055617334322595279, 9.098988733596959751106493271884, 10.26108211508858744484875648665, 11.57582272677838627292886785595
