| L(s) = 1 | − 20.5·2-s − 81·3-s − 90.7·4-s − 622.·5-s + 1.66e3·6-s + 1.39e3·7-s + 1.23e4·8-s + 6.56e3·9-s + 1.27e4·10-s + 8.19e4·11-s + 7.35e3·12-s + 2.21e4·13-s − 2.86e4·14-s + 5.04e4·15-s − 2.07e5·16-s + 5.12e5·17-s − 1.34e5·18-s + 8.50e5·19-s + 5.65e4·20-s − 1.12e5·21-s − 1.68e6·22-s + 1.92e6·23-s − 1.00e6·24-s − 1.56e6·25-s − 4.54e5·26-s − 5.31e5·27-s − 1.26e5·28-s + ⋯ |
| L(s) = 1 | − 0.907·2-s − 0.577·3-s − 0.177·4-s − 0.445·5-s + 0.523·6-s + 0.219·7-s + 1.06·8-s + 0.333·9-s + 0.404·10-s + 1.68·11-s + 0.102·12-s + 0.215·13-s − 0.199·14-s + 0.257·15-s − 0.791·16-s + 1.48·17-s − 0.302·18-s + 1.49·19-s + 0.0790·20-s − 0.126·21-s − 1.53·22-s + 1.43·23-s − 0.616·24-s − 0.801·25-s − 0.195·26-s − 0.192·27-s − 0.0389·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.280626872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.280626872\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 59 | \( 1 - 1.21e7T \) |
| good | 2 | \( 1 + 20.5T + 512T^{2} \) |
| 5 | \( 1 + 622.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.39e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 8.19e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 2.21e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 5.12e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.50e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.92e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.01e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.43e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.61e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.86e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.17e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 7.59e5T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.97e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 7.60e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.31e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.17e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.17e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.20e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.88e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.00e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 6.18e7T + 7.60e17T^{2} \) |
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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
−11.08136419727497989790874145264, −9.710629663248368143912432584375, −9.260544400367516546975997280657, −7.934183774039183421266329616433, −7.20940384431717389806383427476, −5.82816791249466328034125926526, −4.58460102713239238596514345432, −3.50092625427860649631242747377, −1.30143778380932540730290226139, −0.818277480529402993687567937664,
0.818277480529402993687567937664, 1.30143778380932540730290226139, 3.50092625427860649631242747377, 4.58460102713239238596514345432, 5.82816791249466328034125926526, 7.20940384431717389806383427476, 7.934183774039183421266329616433, 9.260544400367516546975997280657, 9.710629663248368143912432584375, 11.08136419727497989790874145264
