| L(s) = 1 | − 849.·2-s − 2.99e3·3-s + 1.97e5·4-s + 3.98e6·5-s + 2.54e6·6-s − 1.37e8·7-s + 2.77e8·8-s − 1.15e9·9-s − 3.38e9·10-s + 7.54e9·11-s − 5.90e8·12-s + 6.89e9·13-s + 1.17e11·14-s − 1.19e10·15-s − 3.39e11·16-s + 4.82e11·17-s + 9.79e11·18-s − 1.83e11·19-s + 7.86e11·20-s + 4.13e11·21-s − 6.41e12·22-s − 9.91e12·23-s − 8.31e11·24-s − 3.21e12·25-s − 5.86e12·26-s + 6.93e12·27-s − 2.72e13·28-s + ⋯ |
| L(s) = 1 | − 1.17·2-s − 0.0878·3-s + 0.376·4-s + 0.911·5-s + 0.103·6-s − 1.29·7-s + 0.731·8-s − 0.992·9-s − 1.06·10-s + 0.965·11-s − 0.0330·12-s + 0.180·13-s + 1.51·14-s − 0.0800·15-s − 1.23·16-s + 0.987·17-s + 1.16·18-s − 0.130·19-s + 0.343·20-s + 0.113·21-s − 1.13·22-s − 1.14·23-s − 0.0642·24-s − 0.168·25-s − 0.211·26-s + 0.174·27-s − 0.486·28-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 | \( 1 + 849.T + 5.24e5T^{2} \) |
| 3 | \( 1 + 2.99e3T + 1.16e9T^{2} \) |
| 5 | \( 1 - 3.98e6T + 1.90e13T^{2} \) |
| 7 | \( 1 + 1.37e8T + 1.13e16T^{2} \) |
| 11 | \( 1 - 7.54e9T + 6.11e19T^{2} \) |
| 13 | \( 1 - 6.89e9T + 1.46e21T^{2} \) |
| 17 | \( 1 - 4.82e11T + 2.39e23T^{2} \) |
| 19 | \( 1 + 1.83e11T + 1.97e24T^{2} \) |
| 23 | \( 1 + 9.91e12T + 7.46e25T^{2} \) |
| 29 | \( 1 - 6.02e13T + 6.10e27T^{2} \) |
| 31 | \( 1 + 2.43e14T + 2.16e28T^{2} \) |
| 37 | \( 1 - 1.46e15T + 6.24e29T^{2} \) |
| 41 | \( 1 + 2.00e15T + 4.39e30T^{2} \) |
| 43 | \( 1 - 5.85e15T + 1.08e31T^{2} \) |
| 53 | \( 1 - 2.63e15T + 5.77e32T^{2} \) |
| 59 | \( 1 + 5.46e16T + 4.42e33T^{2} \) |
| 61 | \( 1 - 2.89e15T + 8.34e33T^{2} \) |
| 67 | \( 1 - 3.43e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 4.77e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 2.85e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 1.47e18T + 1.13e36T^{2} \) |
| 83 | \( 1 - 3.20e18T + 2.90e36T^{2} \) |
| 89 | \( 1 + 5.83e18T + 1.09e37T^{2} \) |
| 97 | \( 1 - 1.34e18T + 5.60e37T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.85628780376199305973556638222, −9.660229179788849009159157696404, −9.319369183791232800942398641682, −8.023646832086530050762550892997, −6.54491001634865161555680305481, −5.69153688790686677776658445896, −3.76968513733445328148221736814, −2.34395008794105508708464011153, −1.06227904799503313914833889579, 0,
1.06227904799503313914833889579, 2.34395008794105508708464011153, 3.76968513733445328148221736814, 5.69153688790686677776658445896, 6.54491001634865161555680305481, 8.023646832086530050762550892997, 9.319369183791232800942398641682, 9.660229179788849009159157696404, 10.85628780376199305973556638222
