| L(s) = 1 | + 364.·2-s + 5.71e4·3-s − 3.91e5·4-s + 3.80e6·5-s + 2.08e7·6-s + 1.73e8·7-s − 3.34e8·8-s + 2.10e9·9-s + 1.38e9·10-s − 1.44e10·11-s − 2.23e10·12-s − 3.07e10·13-s + 6.34e10·14-s + 2.17e11·15-s + 8.31e10·16-s + 5.12e11·17-s + 7.69e11·18-s + 1.95e12·19-s − 1.48e12·20-s + 9.94e12·21-s − 5.28e12·22-s + 5.53e12·23-s − 1.91e13·24-s − 4.60e12·25-s − 1.12e13·26-s + 5.41e13·27-s − 6.80e13·28-s + ⋯ |
| L(s) = 1 | + 0.503·2-s + 1.67·3-s − 0.746·4-s + 0.870·5-s + 0.845·6-s + 1.62·7-s − 0.879·8-s + 1.81·9-s + 0.438·10-s − 1.85·11-s − 1.25·12-s − 0.804·13-s + 0.821·14-s + 1.46·15-s + 0.302·16-s + 1.04·17-s + 0.914·18-s + 1.38·19-s − 0.649·20-s + 2.73·21-s − 0.933·22-s + 0.640·23-s − 1.47·24-s − 0.241·25-s − 0.405·26-s + 1.36·27-s − 1.21·28-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
| \(L(10)\) |
\(\approx\) |
\(6.286323454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.286323454\) |
| \(L(\frac{21}{2})\) |
|
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 | \( 1 - 364.T + 5.24e5T^{2} \) |
| 3 | \( 1 - 5.71e4T + 1.16e9T^{2} \) |
| 5 | \( 1 - 3.80e6T + 1.90e13T^{2} \) |
| 7 | \( 1 - 1.73e8T + 1.13e16T^{2} \) |
| 11 | \( 1 + 1.44e10T + 6.11e19T^{2} \) |
| 13 | \( 1 + 3.07e10T + 1.46e21T^{2} \) |
| 17 | \( 1 - 5.12e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 1.95e12T + 1.97e24T^{2} \) |
| 23 | \( 1 - 5.53e12T + 7.46e25T^{2} \) |
| 29 | \( 1 - 1.10e14T + 6.10e27T^{2} \) |
| 31 | \( 1 - 1.76e14T + 2.16e28T^{2} \) |
| 37 | \( 1 + 5.18e14T + 6.24e29T^{2} \) |
| 41 | \( 1 - 1.89e15T + 4.39e30T^{2} \) |
| 43 | \( 1 + 7.42e14T + 1.08e31T^{2} \) |
| 53 | \( 1 + 3.09e16T + 5.77e32T^{2} \) |
| 59 | \( 1 - 9.94e16T + 4.42e33T^{2} \) |
| 61 | \( 1 + 1.57e17T + 8.34e33T^{2} \) |
| 67 | \( 1 + 9.16e16T + 4.95e34T^{2} \) |
| 71 | \( 1 - 5.70e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 9.48e16T + 2.53e35T^{2} \) |
| 79 | \( 1 - 6.29e17T + 1.13e36T^{2} \) |
| 83 | \( 1 - 2.01e18T + 2.90e36T^{2} \) |
| 89 | \( 1 - 4.26e18T + 1.09e37T^{2} \) |
| 97 | \( 1 + 5.61e18T + 5.60e37T^{2} \) |
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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
−12.24884960547743385929504455053, −10.27073427570345668601960277875, −9.449107123253649170900138745901, −8.169059043240102063702238904525, −7.74231241094850615201592184012, −5.31249871159817018475750876189, −4.75340678739104846765078177659, −3.11193743954771599869641166983, −2.37567438571047277314540328753, −1.11063902985929970581937786803,
1.11063902985929970581937786803, 2.37567438571047277314540328753, 3.11193743954771599869641166983, 4.75340678739104846765078177659, 5.31249871159817018475750876189, 7.74231241094850615201592184012, 8.169059043240102063702238904525, 9.449107123253649170900138745901, 10.27073427570345668601960277875, 12.24884960547743385929504455053
