| L(s) = 1 | − 270.·2-s − 5.98e4·3-s − 4.51e5·4-s − 3.81e6·5-s + 1.61e7·6-s − 1.78e7·7-s + 2.63e8·8-s + 2.41e9·9-s + 1.03e9·10-s + 1.01e10·11-s + 2.69e10·12-s + 4.77e10·13-s + 4.83e9·14-s + 2.27e11·15-s + 1.65e11·16-s − 8.00e11·17-s − 6.53e11·18-s + 7.31e11·19-s + 1.71e12·20-s + 1.06e12·21-s − 2.74e12·22-s + 3.14e11·23-s − 1.57e13·24-s − 4.54e12·25-s − 1.29e13·26-s − 7.48e13·27-s + 8.05e12·28-s + ⋯ |
| L(s) = 1 | − 0.373·2-s − 1.75·3-s − 0.860·4-s − 0.872·5-s + 0.655·6-s − 0.167·7-s + 0.695·8-s + 2.07·9-s + 0.326·10-s + 1.29·11-s + 1.50·12-s + 1.24·13-s + 0.0624·14-s + 1.53·15-s + 0.600·16-s − 1.63·17-s − 0.776·18-s + 0.520·19-s + 0.750·20-s + 0.293·21-s − 0.485·22-s + 0.0363·23-s − 1.21·24-s − 0.238·25-s − 0.466·26-s − 1.88·27-s + 0.143·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(0.5778214858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5778214858\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 47 | \( 1 + 1.11e15T \) |
| good | 2 | \( 1 + 270.T + 5.24e5T^{2} \) |
| 3 | \( 1 + 5.98e4T + 1.16e9T^{2} \) |
| 5 | \( 1 + 3.81e6T + 1.90e13T^{2} \) |
| 7 | \( 1 + 1.78e7T + 1.13e16T^{2} \) |
| 11 | \( 1 - 1.01e10T + 6.11e19T^{2} \) |
| 13 | \( 1 - 4.77e10T + 1.46e21T^{2} \) |
| 17 | \( 1 + 8.00e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 7.31e11T + 1.97e24T^{2} \) |
| 23 | \( 1 - 3.14e11T + 7.46e25T^{2} \) |
| 29 | \( 1 - 4.79e13T + 6.10e27T^{2} \) |
| 31 | \( 1 - 1.56e14T + 2.16e28T^{2} \) |
| 37 | \( 1 - 5.47e14T + 6.24e29T^{2} \) |
| 41 | \( 1 - 3.67e14T + 4.39e30T^{2} \) |
| 43 | \( 1 - 4.79e15T + 1.08e31T^{2} \) |
| 53 | \( 1 - 2.50e16T + 5.77e32T^{2} \) |
| 59 | \( 1 + 1.11e17T + 4.42e33T^{2} \) |
| 61 | \( 1 - 1.36e17T + 8.34e33T^{2} \) |
| 67 | \( 1 - 3.61e17T + 4.95e34T^{2} \) |
| 71 | \( 1 + 2.20e16T + 1.49e35T^{2} \) |
| 73 | \( 1 - 6.10e16T + 2.53e35T^{2} \) |
| 79 | \( 1 + 6.37e17T + 1.13e36T^{2} \) |
| 83 | \( 1 + 1.22e18T + 2.90e36T^{2} \) |
| 89 | \( 1 + 3.86e18T + 1.09e37T^{2} \) |
| 97 | \( 1 + 5.80e18T + 5.60e37T^{2} \) |
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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
−11.56619885551323785060509754414, −10.97079030993008230626912121570, −9.625130463940294268033688229334, −8.423072217735504796740379854628, −6.90345088487210951937040319937, −5.96601157760096937830922462631, −4.52397754994164479033001665996, −3.95457935902653347918651107053, −1.21477394945994472662219663777, −0.51831344796346031015125430106,
0.51831344796346031015125430106, 1.21477394945994472662219663777, 3.95457935902653347918651107053, 4.52397754994164479033001665996, 5.96601157760096937830922462631, 6.90345088487210951937040319937, 8.423072217735504796740379854628, 9.625130463940294268033688229334, 10.97079030993008230626912121570, 11.56619885551323785060509754414
