| L(s) = 1 | + 12.9·2-s − 5.44e4·3-s − 5.24e5·4-s + 3.94e6·5-s − 7.06e5·6-s − 4.65e7·7-s − 1.35e7·8-s + 1.80e9·9-s + 5.11e7·10-s − 9.05e9·11-s + 2.85e10·12-s − 5.30e10·13-s − 6.03e8·14-s − 2.15e11·15-s + 2.74e11·16-s − 3.24e11·17-s + 2.34e10·18-s + 6.05e11·19-s − 2.06e12·20-s + 2.53e12·21-s − 1.17e11·22-s + 9.33e12·23-s + 7.40e11·24-s − 3.48e12·25-s − 6.87e11·26-s − 3.51e13·27-s + 2.44e13·28-s + ⋯ |
| L(s) = 1 | + 0.0179·2-s − 1.59·3-s − 0.999·4-s + 0.904·5-s − 0.0286·6-s − 0.436·7-s − 0.0357·8-s + 1.55·9-s + 0.0161·10-s − 1.15·11-s + 1.59·12-s − 1.38·13-s − 0.00780·14-s − 1.44·15-s + 0.999·16-s − 0.662·17-s + 0.0278·18-s + 0.430·19-s − 0.903·20-s + 0.697·21-s − 0.0207·22-s + 1.08·23-s + 0.0572·24-s − 0.182·25-s − 0.0248·26-s − 0.886·27-s + 0.436·28-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 | \( 1 - 12.9T + 5.24e5T^{2} \) |
| 3 | \( 1 + 5.44e4T + 1.16e9T^{2} \) |
| 5 | \( 1 - 3.94e6T + 1.90e13T^{2} \) |
| 7 | \( 1 + 4.65e7T + 1.13e16T^{2} \) |
| 11 | \( 1 + 9.05e9T + 6.11e19T^{2} \) |
| 13 | \( 1 + 5.30e10T + 1.46e21T^{2} \) |
| 17 | \( 1 + 3.24e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 6.05e11T + 1.97e24T^{2} \) |
| 23 | \( 1 - 9.33e12T + 7.46e25T^{2} \) |
| 29 | \( 1 - 6.14e13T + 6.10e27T^{2} \) |
| 31 | \( 1 + 1.23e14T + 2.16e28T^{2} \) |
| 37 | \( 1 - 1.57e15T + 6.24e29T^{2} \) |
| 41 | \( 1 - 2.84e15T + 4.39e30T^{2} \) |
| 43 | \( 1 - 4.23e15T + 1.08e31T^{2} \) |
| 53 | \( 1 + 4.13e16T + 5.77e32T^{2} \) |
| 59 | \( 1 - 1.01e17T + 4.42e33T^{2} \) |
| 61 | \( 1 - 1.85e16T + 8.34e33T^{2} \) |
| 67 | \( 1 + 9.95e16T + 4.95e34T^{2} \) |
| 71 | \( 1 - 4.77e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 4.48e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 3.87e17T + 1.13e36T^{2} \) |
| 83 | \( 1 + 7.20e17T + 2.90e36T^{2} \) |
| 89 | \( 1 + 3.00e18T + 1.09e37T^{2} \) |
| 97 | \( 1 - 4.57e18T + 5.60e37T^{2} \) |
| 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
−11.15460185114011437537379586653, −10.07927030549641001887391859474, −9.390930037232586804765327827507, −7.51807489852356463419555425711, −6.12777910309537983989031680250, −5.27477210720340973542601196864, −4.57524014907863555094780079344, −2.58206338769439002253025258819, −0.853127467262667652602214932098, 0,
0.853127467262667652602214932098, 2.58206338769439002253025258819, 4.57524014907863555094780079344, 5.27477210720340973542601196864, 6.12777910309537983989031680250, 7.51807489852356463419555425711, 9.390930037232586804765327827507, 10.07927030549641001887391859474, 11.15460185114011437537379586653
