| L(s) = 1 | − 1.32e3·2-s − 5.86e4·3-s + 1.22e6·4-s − 2.69e6·5-s + 7.75e7·6-s − 1.70e8·7-s − 9.25e8·8-s + 2.27e9·9-s + 3.55e9·10-s + 2.84e9·11-s − 7.17e10·12-s − 4.77e9·13-s + 2.25e11·14-s + 1.57e11·15-s + 5.81e11·16-s − 4.56e11·17-s − 3.01e12·18-s − 8.99e11·19-s − 3.29e12·20-s + 9.99e12·21-s − 3.75e12·22-s − 1.55e13·23-s + 5.42e13·24-s − 1.18e13·25-s + 6.31e12·26-s − 6.54e13·27-s − 2.08e14·28-s + ⋯ |
| L(s) = 1 | − 1.82·2-s − 1.72·3-s + 2.33·4-s − 0.616·5-s + 3.14·6-s − 1.59·7-s − 2.43·8-s + 1.95·9-s + 1.12·10-s + 0.363·11-s − 4.01·12-s − 0.124·13-s + 2.91·14-s + 1.06·15-s + 2.11·16-s − 0.934·17-s − 3.57·18-s − 0.639·19-s − 1.43·20-s + 2.74·21-s − 0.663·22-s − 1.79·23-s + 4.19·24-s − 0.620·25-s + 0.228·26-s − 1.65·27-s − 3.72·28-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 | \( 1 + 1.32e3T + 5.24e5T^{2} \) |
| 3 | \( 1 + 5.86e4T + 1.16e9T^{2} \) |
| 5 | \( 1 + 2.69e6T + 1.90e13T^{2} \) |
| 7 | \( 1 + 1.70e8T + 1.13e16T^{2} \) |
| 11 | \( 1 - 2.84e9T + 6.11e19T^{2} \) |
| 13 | \( 1 + 4.77e9T + 1.46e21T^{2} \) |
| 17 | \( 1 + 4.56e11T + 2.39e23T^{2} \) |
| 19 | \( 1 + 8.99e11T + 1.97e24T^{2} \) |
| 23 | \( 1 + 1.55e13T + 7.46e25T^{2} \) |
| 29 | \( 1 + 9.23e13T + 6.10e27T^{2} \) |
| 31 | \( 1 + 6.23e13T + 2.16e28T^{2} \) |
| 37 | \( 1 - 2.05e14T + 6.24e29T^{2} \) |
| 41 | \( 1 - 2.83e15T + 4.39e30T^{2} \) |
| 43 | \( 1 + 1.74e15T + 1.08e31T^{2} \) |
| 53 | \( 1 + 1.80e16T + 5.77e32T^{2} \) |
| 59 | \( 1 - 6.22e16T + 4.42e33T^{2} \) |
| 61 | \( 1 + 1.45e17T + 8.34e33T^{2} \) |
| 67 | \( 1 - 3.92e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 5.02e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 9.66e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 1.46e18T + 1.13e36T^{2} \) |
| 83 | \( 1 - 5.27e17T + 2.90e36T^{2} \) |
| 89 | \( 1 - 2.55e18T + 1.09e37T^{2} \) |
| 97 | \( 1 - 1.02e19T + 5.60e37T^{2} \) |
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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
−11.03772963989041123143774017859, −10.09761999698662838625024061023, −9.264849785984288305817894264675, −7.65405148908642454909390843187, −6.58437967825542023045622586544, −6.03620253927588978566816700798, −3.92915223249344623982697943898, −2.01932820446251508356210627687, −0.51832372821480489024273819333, 0,
0.51832372821480489024273819333, 2.01932820446251508356210627687, 3.92915223249344623982697943898, 6.03620253927588978566816700798, 6.58437967825542023045622586544, 7.65405148908642454909390843187, 9.264849785984288305817894264675, 10.09761999698662838625024061023, 11.03772963989041123143774017859
