L(s) = 1 | − 2.61·2-s + 3-s + 4.85·4-s − 3·5-s − 2.61·6-s − 2.38·7-s − 7.47·8-s + 9-s + 7.85·10-s + 3.47·11-s + 4.85·12-s − 6.70·13-s + 6.23·14-s − 3·15-s + 9.85·16-s − 2.61·17-s − 2.61·18-s − 5.85·19-s − 14.5·20-s − 2.38·21-s − 9.09·22-s − 1.38·23-s − 7.47·24-s + 4·25-s + 17.5·26-s + 27-s − 11.5·28-s + ⋯ |
L(s) = 1 | − 1.85·2-s + 0.577·3-s + 2.42·4-s − 1.34·5-s − 1.06·6-s − 0.900·7-s − 2.64·8-s + 0.333·9-s + 2.48·10-s + 1.04·11-s + 1.40·12-s − 1.86·13-s + 1.66·14-s − 0.774·15-s + 2.46·16-s − 0.634·17-s − 0.617·18-s − 1.34·19-s − 3.25·20-s − 0.519·21-s − 1.93·22-s − 0.288·23-s − 1.52·24-s + 0.800·25-s + 3.44·26-s + 0.192·27-s − 2.18·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 59 | \( 1 - T \) |
good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 5 | \( 1 + 3T + 5T^{2} \) |
| 7 | \( 1 + 2.38T + 7T^{2} \) |
| 11 | \( 1 - 3.47T + 11T^{2} \) |
| 13 | \( 1 + 6.70T + 13T^{2} \) |
| 17 | \( 1 + 2.61T + 17T^{2} \) |
| 19 | \( 1 + 5.85T + 19T^{2} \) |
| 23 | \( 1 + 1.38T + 23T^{2} \) |
| 29 | \( 1 + 4.38T + 29T^{2} \) |
| 31 | \( 1 + 2.38T + 31T^{2} \) |
| 37 | \( 1 - 5.09T + 37T^{2} \) |
| 41 | \( 1 + 6.09T + 41T^{2} \) |
| 43 | \( 1 - 12.7T + 43T^{2} \) |
| 47 | \( 1 + 4.85T + 47T^{2} \) |
| 53 | \( 1 - 5.47T + 53T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 + 4.23T + 67T^{2} \) |
| 71 | \( 1 - 4.23T + 71T^{2} \) |
| 73 | \( 1 - 8.09T + 73T^{2} \) |
| 79 | \( 1 + 3T + 79T^{2} \) |
| 83 | \( 1 + 1.14T + 83T^{2} \) |
| 89 | \( 1 - 6.09T + 89T^{2} \) |
| 97 | \( 1 + 9.47T + 97T^{2} \) |
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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
−11.92696214469704744571880174187, −10.99184770821395818272835611046, −9.827331182086964785223470418324, −9.180492390656979253391309638881, −8.221589439921848690314643683124, −7.31829646127061690907670913077, −6.62774106369167144361747897254, −3.98321240869532954503946024592, −2.41602178571510672764464498026, 0,
2.41602178571510672764464498026, 3.98321240869532954503946024592, 6.62774106369167144361747897254, 7.31829646127061690907670913077, 8.221589439921848690314643683124, 9.180492390656979253391309638881, 9.827331182086964785223470418324, 10.99184770821395818272835611046, 11.92696214469704744571880174187