| L(s) = 1 | + 875.·2-s + 7.01e3·3-s + 2.42e5·4-s + 5.09e6·5-s + 6.13e6·6-s + 5.83e7·7-s − 2.46e8·8-s − 1.11e9·9-s + 4.45e9·10-s + 9.46e9·11-s + 1.70e9·12-s + 4.52e10·13-s + 5.10e10·14-s + 3.57e10·15-s − 3.43e11·16-s − 3.27e11·17-s − 9.74e11·18-s + 7.22e11·19-s + 1.23e12·20-s + 4.09e11·21-s + 8.28e12·22-s + 4.75e12·23-s − 1.73e12·24-s + 6.85e12·25-s + 3.96e13·26-s − 1.59e13·27-s + 1.41e13·28-s + ⋯ |
| L(s) = 1 | + 1.20·2-s + 0.205·3-s + 0.462·4-s + 1.16·5-s + 0.248·6-s + 0.546·7-s − 0.650·8-s − 0.957·9-s + 1.40·10-s + 1.21·11-s + 0.0951·12-s + 1.18·13-s + 0.660·14-s + 0.239·15-s − 1.24·16-s − 0.670·17-s − 1.15·18-s + 0.513·19-s + 0.539·20-s + 0.112·21-s + 1.46·22-s + 0.550·23-s − 0.133·24-s + 0.359·25-s + 1.43·26-s − 0.402·27-s + 0.252·28-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
| \(L(10)\) |
\(\approx\) |
\(6.202527790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.202527790\) |
| \(L(\frac{21}{2})\) |
|
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 | \( 1 - 875.T + 5.24e5T^{2} \) |
| 3 | \( 1 - 7.01e3T + 1.16e9T^{2} \) |
| 5 | \( 1 - 5.09e6T + 1.90e13T^{2} \) |
| 7 | \( 1 - 5.83e7T + 1.13e16T^{2} \) |
| 11 | \( 1 - 9.46e9T + 6.11e19T^{2} \) |
| 13 | \( 1 - 4.52e10T + 1.46e21T^{2} \) |
| 17 | \( 1 + 3.27e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 7.22e11T + 1.97e24T^{2} \) |
| 23 | \( 1 - 4.75e12T + 7.46e25T^{2} \) |
| 29 | \( 1 - 7.23e13T + 6.10e27T^{2} \) |
| 31 | \( 1 - 1.77e14T + 2.16e28T^{2} \) |
| 37 | \( 1 + 7.62e14T + 6.24e29T^{2} \) |
| 41 | \( 1 - 1.45e15T + 4.39e30T^{2} \) |
| 43 | \( 1 - 6.03e15T + 1.08e31T^{2} \) |
| 53 | \( 1 + 2.69e16T + 5.77e32T^{2} \) |
| 59 | \( 1 - 1.08e17T + 4.42e33T^{2} \) |
| 61 | \( 1 - 1.41e16T + 8.34e33T^{2} \) |
| 67 | \( 1 + 2.74e17T + 4.95e34T^{2} \) |
| 71 | \( 1 + 8.65e15T + 1.49e35T^{2} \) |
| 73 | \( 1 - 2.01e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 8.23e17T + 1.13e36T^{2} \) |
| 83 | \( 1 - 1.94e18T + 2.90e36T^{2} \) |
| 89 | \( 1 + 2.61e18T + 1.09e37T^{2} \) |
| 97 | \( 1 - 1.47e19T + 5.60e37T^{2} \) |
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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
−11.97639059709312197644583055481, −11.03943522937446012502278291090, −9.346908610898168508520948815188, −8.572096690901275824924221097889, −6.47592463599451904688232617922, −5.82685477217923415194677458326, −4.67641605728137889066989491230, −3.45287987508113784719146867020, −2.32354269444402193726267617973, −1.04389655882486079458241811955,
1.04389655882486079458241811955, 2.32354269444402193726267617973, 3.45287987508113784719146867020, 4.67641605728137889066989491230, 5.82685477217923415194677458326, 6.47592463599451904688232617922, 8.572096690901275824924221097889, 9.346908610898168508520948815188, 11.03943522937446012502278291090, 11.97639059709312197644583055481
