| L(s) = 1 | + 265.·2-s − 2.15e4·3-s − 4.53e5·4-s − 4.39e5·5-s − 5.71e6·6-s − 1.15e8·7-s − 2.59e8·8-s − 6.99e8·9-s − 1.16e8·10-s − 5.04e8·11-s + 9.75e9·12-s + 4.52e10·13-s − 3.06e10·14-s + 9.45e9·15-s + 1.68e11·16-s + 4.63e11·17-s − 1.86e11·18-s + 2.78e12·19-s + 1.99e11·20-s + 2.48e12·21-s − 1.34e11·22-s − 9.27e12·23-s + 5.59e12·24-s − 1.88e13·25-s + 1.20e13·26-s + 4.00e13·27-s + 5.23e13·28-s + ⋯ |
| L(s) = 1 | + 0.367·2-s − 0.630·3-s − 0.865·4-s − 0.100·5-s − 0.231·6-s − 1.08·7-s − 0.684·8-s − 0.602·9-s − 0.0369·10-s − 0.0645·11-s + 0.545·12-s + 1.18·13-s − 0.396·14-s + 0.0634·15-s + 0.613·16-s + 0.946·17-s − 0.221·18-s + 1.98·19-s + 0.0870·20-s + 0.681·21-s − 0.0236·22-s − 1.07·23-s + 0.431·24-s − 0.989·25-s + 0.434·26-s + 1.01·27-s + 0.935·28-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
| \(L(10)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{21}{2})\) |
|
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 | \( 1 - 265.T + 5.24e5T^{2} \) |
| 3 | \( 1 + 2.15e4T + 1.16e9T^{2} \) |
| 5 | \( 1 + 4.39e5T + 1.90e13T^{2} \) |
| 7 | \( 1 + 1.15e8T + 1.13e16T^{2} \) |
| 11 | \( 1 + 5.04e8T + 6.11e19T^{2} \) |
| 13 | \( 1 - 4.52e10T + 1.46e21T^{2} \) |
| 17 | \( 1 - 4.63e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 2.78e12T + 1.97e24T^{2} \) |
| 23 | \( 1 + 9.27e12T + 7.46e25T^{2} \) |
| 29 | \( 1 + 7.18e13T + 6.10e27T^{2} \) |
| 31 | \( 1 + 4.58e13T + 2.16e28T^{2} \) |
| 37 | \( 1 - 1.17e14T + 6.24e29T^{2} \) |
| 41 | \( 1 - 1.84e15T + 4.39e30T^{2} \) |
| 43 | \( 1 - 3.70e15T + 1.08e31T^{2} \) |
| 53 | \( 1 - 1.48e16T + 5.77e32T^{2} \) |
| 59 | \( 1 - 3.24e16T + 4.42e33T^{2} \) |
| 61 | \( 1 + 3.74e16T + 8.34e33T^{2} \) |
| 67 | \( 1 + 3.34e17T + 4.95e34T^{2} \) |
| 71 | \( 1 + 9.90e15T + 1.49e35T^{2} \) |
| 73 | \( 1 - 4.65e17T + 2.53e35T^{2} \) |
| 79 | \( 1 + 1.23e18T + 1.13e36T^{2} \) |
| 83 | \( 1 + 1.04e18T + 2.90e36T^{2} \) |
| 89 | \( 1 - 6.35e17T + 1.09e37T^{2} \) |
| 97 | \( 1 + 1.32e18T + 5.60e37T^{2} \) |
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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
−11.48710557052865593969530065712, −10.00763508962632990492059136252, −9.112968061657666195329575522136, −7.73044327675758085947743276936, −5.95629443804981368803685812388, −5.57564822277378505029073641801, −3.90404816438394327293861640406, −3.11169997560704590499470743568, −0.973303151256287247515274029143, 0,
0.973303151256287247515274029143, 3.11169997560704590499470743568, 3.90404816438394327293861640406, 5.57564822277378505029073641801, 5.95629443804981368803685812388, 7.73044327675758085947743276936, 9.112968061657666195329575522136, 10.00763508962632990492059136252, 11.48710557052865593969530065712
