| L(s) = 1 | + 952.·2-s + 1.27e4·3-s + 3.82e5·4-s − 4.94e6·5-s + 1.21e7·6-s − 2.00e8·7-s − 1.35e8·8-s − 9.99e8·9-s − 4.70e9·10-s + 7.97e9·11-s + 4.88e9·12-s + 6.54e10·13-s − 1.90e11·14-s − 6.31e10·15-s − 3.29e11·16-s − 1.28e11·17-s − 9.51e11·18-s − 8.96e11·19-s − 1.89e12·20-s − 2.56e12·21-s + 7.59e12·22-s − 1.18e12·23-s − 1.72e12·24-s + 5.36e12·25-s + 6.23e13·26-s − 2.76e13·27-s − 7.66e13·28-s + ⋯ |
| L(s) = 1 | + 1.31·2-s + 0.374·3-s + 0.729·4-s − 1.13·5-s + 0.492·6-s − 1.87·7-s − 0.355·8-s − 0.859·9-s − 1.48·10-s + 1.01·11-s + 0.273·12-s + 1.71·13-s − 2.46·14-s − 0.424·15-s − 1.19·16-s − 0.262·17-s − 1.13·18-s − 0.637·19-s − 0.825·20-s − 0.703·21-s + 1.34·22-s − 0.136·23-s − 0.133·24-s + 0.281·25-s + 2.25·26-s − 0.696·27-s − 1.36·28-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
| \(L(10)\) |
\(\approx\) |
\(2.246727796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.246727796\) |
| \(L(\frac{21}{2})\) |
|
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 | \( 1 - 952.T + 5.24e5T^{2} \) |
| 3 | \( 1 - 1.27e4T + 1.16e9T^{2} \) |
| 5 | \( 1 + 4.94e6T + 1.90e13T^{2} \) |
| 7 | \( 1 + 2.00e8T + 1.13e16T^{2} \) |
| 11 | \( 1 - 7.97e9T + 6.11e19T^{2} \) |
| 13 | \( 1 - 6.54e10T + 1.46e21T^{2} \) |
| 17 | \( 1 + 1.28e11T + 2.39e23T^{2} \) |
| 19 | \( 1 + 8.96e11T + 1.97e24T^{2} \) |
| 23 | \( 1 + 1.18e12T + 7.46e25T^{2} \) |
| 29 | \( 1 - 3.74e13T + 6.10e27T^{2} \) |
| 31 | \( 1 - 1.01e14T + 2.16e28T^{2} \) |
| 37 | \( 1 - 1.42e15T + 6.24e29T^{2} \) |
| 41 | \( 1 - 2.69e15T + 4.39e30T^{2} \) |
| 43 | \( 1 + 6.01e15T + 1.08e31T^{2} \) |
| 53 | \( 1 - 2.59e16T + 5.77e32T^{2} \) |
| 59 | \( 1 + 6.87e16T + 4.42e33T^{2} \) |
| 61 | \( 1 + 3.50e16T + 8.34e33T^{2} \) |
| 67 | \( 1 - 8.63e16T + 4.95e34T^{2} \) |
| 71 | \( 1 - 1.85e17T + 1.49e35T^{2} \) |
| 73 | \( 1 - 6.73e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 3.37e17T + 1.13e36T^{2} \) |
| 83 | \( 1 + 3.65e17T + 2.90e36T^{2} \) |
| 89 | \( 1 + 3.77e18T + 1.09e37T^{2} \) |
| 97 | \( 1 - 1.75e18T + 5.60e37T^{2} \) |
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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
−12.06945299272021857977221695942, −11.18843795525467711539755333525, −9.347602505990064107523218757402, −8.394284800079527035019134579094, −6.54094258091458557744478451203, −6.05779752793175324213257152857, −4.11132717355505114290497724826, −3.60678290692182288022898590631, −2.78418178656139582736824428738, −0.55024432434143588980895098879,
0.55024432434143588980895098879, 2.78418178656139582736824428738, 3.60678290692182288022898590631, 4.11132717355505114290497724826, 6.05779752793175324213257152857, 6.54094258091458557744478451203, 8.394284800079527035019134579094, 9.347602505990064107523218757402, 11.18843795525467711539755333525, 12.06945299272021857977221695942
