| L(s) = 1 | − 461.·2-s − 8.00e3·3-s − 3.11e5·4-s − 9.35e4·5-s + 3.68e6·6-s + 1.17e8·7-s + 3.85e8·8-s − 1.09e9·9-s + 4.31e7·10-s − 1.36e10·11-s + 2.49e9·12-s + 5.04e10·13-s − 5.39e10·14-s + 7.48e8·15-s − 1.43e10·16-s − 9.55e10·17-s + 5.06e11·18-s + 2.06e12·19-s + 2.91e10·20-s − 9.36e11·21-s + 6.30e12·22-s + 1.16e13·23-s − 3.08e12·24-s − 1.90e13·25-s − 2.32e13·26-s + 1.80e13·27-s − 3.65e13·28-s + ⋯ |
| L(s) = 1 | − 0.636·2-s − 0.234·3-s − 0.594·4-s − 0.0214·5-s + 0.149·6-s + 1.09·7-s + 1.01·8-s − 0.944·9-s + 0.0136·10-s − 1.74·11-s + 0.139·12-s + 1.31·13-s − 0.698·14-s + 0.00502·15-s − 0.0520·16-s − 0.195·17-s + 0.601·18-s + 1.46·19-s + 0.0127·20-s − 0.257·21-s + 1.11·22-s + 1.34·23-s − 0.238·24-s − 0.999·25-s − 0.839·26-s + 0.456·27-s − 0.652·28-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
| \(L(10)\) |
\(\approx\) |
\(0.9020970680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9020970680\) |
| \(L(\frac{21}{2})\) |
|
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 | \( 1 + 461.T + 5.24e5T^{2} \) |
| 3 | \( 1 + 8.00e3T + 1.16e9T^{2} \) |
| 5 | \( 1 + 9.35e4T + 1.90e13T^{2} \) |
| 7 | \( 1 - 1.17e8T + 1.13e16T^{2} \) |
| 11 | \( 1 + 1.36e10T + 6.11e19T^{2} \) |
| 13 | \( 1 - 5.04e10T + 1.46e21T^{2} \) |
| 17 | \( 1 + 9.55e10T + 2.39e23T^{2} \) |
| 19 | \( 1 - 2.06e12T + 1.97e24T^{2} \) |
| 23 | \( 1 - 1.16e13T + 7.46e25T^{2} \) |
| 29 | \( 1 + 1.38e14T + 6.10e27T^{2} \) |
| 31 | \( 1 + 3.67e12T + 2.16e28T^{2} \) |
| 37 | \( 1 - 9.11e14T + 6.24e29T^{2} \) |
| 41 | \( 1 + 3.88e15T + 4.39e30T^{2} \) |
| 43 | \( 1 - 9.66e14T + 1.08e31T^{2} \) |
| 53 | \( 1 + 3.76e16T + 5.77e32T^{2} \) |
| 59 | \( 1 + 1.66e16T + 4.42e33T^{2} \) |
| 61 | \( 1 + 7.31e16T + 8.34e33T^{2} \) |
| 67 | \( 1 - 3.20e17T + 4.95e34T^{2} \) |
| 71 | \( 1 + 5.72e17T + 1.49e35T^{2} \) |
| 73 | \( 1 - 2.07e14T + 2.53e35T^{2} \) |
| 79 | \( 1 - 1.72e18T + 1.13e36T^{2} \) |
| 83 | \( 1 - 2.00e18T + 2.90e36T^{2} \) |
| 89 | \( 1 + 2.84e18T + 1.09e37T^{2} \) |
| 97 | \( 1 - 2.04e18T + 5.60e37T^{2} \) |
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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
−11.35510452660617657442402012997, −10.81687831290133243624372974681, −9.390967778095983194803512679890, −8.265493403785806922189823186575, −7.66839043550156654987604418807, −5.59632037634471623527106737155, −4.91663037809410062447501261937, −3.28352042895113835429461656090, −1.68248290347336098146072972817, −0.52182152775848337035201633752,
0.52182152775848337035201633752, 1.68248290347336098146072972817, 3.28352042895113835429461656090, 4.91663037809410062447501261937, 5.59632037634471623527106737155, 7.66839043550156654987604418807, 8.265493403785806922189823186575, 9.390967778095983194803512679890, 10.81687831290133243624372974681, 11.35510452660617657442402012997
