| L(s) = 1 | + 43.4·2-s − 2.44e4·3-s − 5.22e5·4-s − 8.30e6·5-s − 1.06e6·6-s − 4.43e7·7-s − 4.54e7·8-s − 5.63e8·9-s − 3.60e8·10-s − 9.46e9·11-s + 1.27e10·12-s + 3.66e10·13-s − 1.92e9·14-s + 2.03e11·15-s + 2.71e11·16-s + 1.13e10·17-s − 2.44e10·18-s − 2.05e12·19-s + 4.33e12·20-s + 1.08e12·21-s − 4.11e11·22-s − 4.07e12·23-s + 1.11e12·24-s + 4.98e13·25-s + 1.59e12·26-s + 4.22e13·27-s + 2.31e13·28-s + ⋯ |
| L(s) = 1 | + 0.0599·2-s − 0.717·3-s − 0.996·4-s − 1.90·5-s − 0.0430·6-s − 0.415·7-s − 0.119·8-s − 0.484·9-s − 0.113·10-s − 1.21·11-s + 0.715·12-s + 0.958·13-s − 0.0249·14-s + 1.36·15-s + 0.989·16-s + 0.0232·17-s − 0.0290·18-s − 1.45·19-s + 1.89·20-s + 0.298·21-s − 0.0725·22-s − 0.471·23-s + 0.0859·24-s + 2.61·25-s + 0.0574·26-s + 1.06·27-s + 0.414·28-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 | \( 1 - 43.4T + 5.24e5T^{2} \) |
| 3 | \( 1 + 2.44e4T + 1.16e9T^{2} \) |
| 5 | \( 1 + 8.30e6T + 1.90e13T^{2} \) |
| 7 | \( 1 + 4.43e7T + 1.13e16T^{2} \) |
| 11 | \( 1 + 9.46e9T + 6.11e19T^{2} \) |
| 13 | \( 1 - 3.66e10T + 1.46e21T^{2} \) |
| 17 | \( 1 - 1.13e10T + 2.39e23T^{2} \) |
| 19 | \( 1 + 2.05e12T + 1.97e24T^{2} \) |
| 23 | \( 1 + 4.07e12T + 7.46e25T^{2} \) |
| 29 | \( 1 - 5.46e13T + 6.10e27T^{2} \) |
| 31 | \( 1 - 5.89e13T + 2.16e28T^{2} \) |
| 37 | \( 1 - 4.23e14T + 6.24e29T^{2} \) |
| 41 | \( 1 + 6.18e14T + 4.39e30T^{2} \) |
| 43 | \( 1 + 3.47e15T + 1.08e31T^{2} \) |
| 53 | \( 1 + 2.51e16T + 5.77e32T^{2} \) |
| 59 | \( 1 - 8.07e16T + 4.42e33T^{2} \) |
| 61 | \( 1 - 6.54e16T + 8.34e33T^{2} \) |
| 67 | \( 1 - 1.17e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 1.21e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 7.34e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 1.25e17T + 1.13e36T^{2} \) |
| 83 | \( 1 - 2.59e18T + 2.90e36T^{2} \) |
| 89 | \( 1 - 5.55e17T + 1.09e37T^{2} \) |
| 97 | \( 1 + 6.17e17T + 5.60e37T^{2} \) |
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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
−11.30694400677462953041047847443, −10.38833618560159596788077698247, −8.530731446145586168788011902965, −8.071415340468470879890414630073, −6.42347125238897046614441285850, −5.04897467947203081601459990905, −4.10640079524020490452459133766, −3.08873999953625520525162681693, −0.62690538019584302170473287841, 0,
0.62690538019584302170473287841, 3.08873999953625520525162681693, 4.10640079524020490452459133766, 5.04897467947203081601459990905, 6.42347125238897046614441285850, 8.071415340468470879890414630073, 8.530731446145586168788011902965, 10.38833618560159596788077698247, 11.30694400677462953041047847443
