| L(s) = 1 | − 22.5·2-s − 82.1·3-s − 1.26·4-s + 454.·5-s + 1.85e3·6-s − 6.48e3·7-s + 1.15e4·8-s − 1.29e4·9-s − 1.02e4·10-s − 7.49e4·11-s + 104.·12-s − 4.20e4·13-s + 1.46e5·14-s − 3.73e4·15-s − 2.61e5·16-s − 5.65e5·17-s + 2.92e5·18-s + 1.80e5·19-s − 577.·20-s + 5.32e5·21-s + 1.69e6·22-s + 6.34e5·23-s − 9.52e5·24-s − 1.74e6·25-s + 9.50e5·26-s + 2.67e6·27-s + 8.21e3·28-s + ⋯ |
| L(s) = 1 | − 0.998·2-s − 0.585·3-s − 0.00247·4-s + 0.325·5-s + 0.584·6-s − 1.02·7-s + 1.00·8-s − 0.657·9-s − 0.325·10-s − 1.54·11-s + 0.00144·12-s − 0.408·13-s + 1.01·14-s − 0.190·15-s − 0.997·16-s − 1.64·17-s + 0.656·18-s + 0.317·19-s − 0.000806·20-s + 0.597·21-s + 1.54·22-s + 0.472·23-s − 0.585·24-s − 0.894·25-s + 0.408·26-s + 0.970·27-s + 0.00252·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.2432887985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2432887985\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 - 3.41e6T \) |
| good | 2 | \( 1 + 22.5T + 512T^{2} \) |
| 3 | \( 1 + 82.1T + 1.96e4T^{2} \) |
| 5 | \( 1 - 454.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 6.48e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 7.49e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 4.20e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 5.65e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.80e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 6.34e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.45e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.98e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 8.43e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.12e7T + 3.27e14T^{2} \) |
| 47 | \( 1 + 5.09e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.01e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.42e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.00e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.03e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.49e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.32e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 6.08e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.28e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.15e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + 5.10e6T + 7.60e17T^{2} \) |
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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
−13.69821187482910536401725522159, −12.88299523672779579284443600375, −11.19401230230700329659814312214, −10.19509101626124551551180254389, −9.196917985082836666937905232772, −7.88291254164746386931333557224, −6.35036434053094817056363142074, −4.88361304843350285086197343714, −2.56406276487276161789573450889, −0.37342985725343957745841034134,
0.37342985725343957745841034134, 2.56406276487276161789573450889, 4.88361304843350285086197343714, 6.35036434053094817056363142074, 7.88291254164746386931333557224, 9.196917985082836666937905232772, 10.19509101626124551551180254389, 11.19401230230700329659814312214, 12.88299523672779579284443600375, 13.69821187482910536401725522159
