| L(s) = 1 | − 1.26e3·2-s − 5.65e3·3-s + 1.07e6·4-s + 3.30e5·5-s + 7.15e6·6-s + 4.35e6·7-s − 6.95e8·8-s − 1.13e9·9-s − 4.18e8·10-s + 4.87e9·11-s − 6.08e9·12-s + 6.53e10·13-s − 5.50e9·14-s − 1.87e9·15-s + 3.16e11·16-s + 9.28e10·17-s + 1.42e12·18-s − 1.18e12·19-s + 3.55e11·20-s − 2.46e10·21-s − 6.16e12·22-s + 5.61e11·23-s + 3.93e12·24-s − 1.89e13·25-s − 8.26e13·26-s + 1.29e13·27-s + 4.67e12·28-s + ⋯ |
| L(s) = 1 | − 1.74·2-s − 0.166·3-s + 2.04·4-s + 0.0757·5-s + 0.289·6-s + 0.0407·7-s − 1.83·8-s − 0.972·9-s − 0.132·10-s + 0.623·11-s − 0.340·12-s + 1.70·13-s − 0.0711·14-s − 0.0125·15-s + 1.15·16-s + 0.189·17-s + 1.69·18-s − 0.845·19-s + 0.155·20-s − 0.00676·21-s − 1.08·22-s + 0.0650·23-s + 0.304·24-s − 0.994·25-s − 2.98·26-s + 0.327·27-s + 0.0835·28-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 | \( 1 + 1.26e3T + 5.24e5T^{2} \) |
| 3 | \( 1 + 5.65e3T + 1.16e9T^{2} \) |
| 5 | \( 1 - 3.30e5T + 1.90e13T^{2} \) |
| 7 | \( 1 - 4.35e6T + 1.13e16T^{2} \) |
| 11 | \( 1 - 4.87e9T + 6.11e19T^{2} \) |
| 13 | \( 1 - 6.53e10T + 1.46e21T^{2} \) |
| 17 | \( 1 - 9.28e10T + 2.39e23T^{2} \) |
| 19 | \( 1 + 1.18e12T + 1.97e24T^{2} \) |
| 23 | \( 1 - 5.61e11T + 7.46e25T^{2} \) |
| 29 | \( 1 + 8.92e13T + 6.10e27T^{2} \) |
| 31 | \( 1 - 9.28e13T + 2.16e28T^{2} \) |
| 37 | \( 1 + 4.48e13T + 6.24e29T^{2} \) |
| 41 | \( 1 + 9.68e14T + 4.39e30T^{2} \) |
| 43 | \( 1 + 4.04e15T + 1.08e31T^{2} \) |
| 53 | \( 1 - 3.97e16T + 5.77e32T^{2} \) |
| 59 | \( 1 - 8.26e16T + 4.42e33T^{2} \) |
| 61 | \( 1 - 1.30e17T + 8.34e33T^{2} \) |
| 67 | \( 1 - 7.04e16T + 4.95e34T^{2} \) |
| 71 | \( 1 + 2.99e17T + 1.49e35T^{2} \) |
| 73 | \( 1 - 5.42e17T + 2.53e35T^{2} \) |
| 79 | \( 1 + 9.66e17T + 1.13e36T^{2} \) |
| 83 | \( 1 + 1.90e17T + 2.90e36T^{2} \) |
| 89 | \( 1 - 8.53e17T + 1.09e37T^{2} \) |
| 97 | \( 1 + 8.97e18T + 5.60e37T^{2} \) |
| 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
−11.09905038337614281139948236337, −9.960624468113138895576870071957, −8.766230242081325423816898420460, −8.242121459491289678864027437985, −6.74533953065226005121235886846, −5.82669941503139335149698585651, −3.64075973260830116665814492577, −2.11481467685104818708458829048, −1.07861962159823530466769472305, 0,
1.07861962159823530466769472305, 2.11481467685104818708458829048, 3.64075973260830116665814492577, 5.82669941503139335149698585651, 6.74533953065226005121235886846, 8.242121459491289678864027437985, 8.766230242081325423816898420460, 9.960624468113138895576870071957, 11.09905038337614281139948236337
