| L(s) = 1 | − 24.8·2-s + 81·3-s + 106.·4-s − 222.·5-s − 2.01e3·6-s − 2.49e3·7-s + 1.00e4·8-s + 6.56e3·9-s + 5.52e3·10-s − 1.77e4·11-s + 8.65e3·12-s − 1.22e4·13-s + 6.20e4·14-s − 1.79e4·15-s − 3.05e5·16-s − 1.70e5·17-s − 1.63e5·18-s + 8.00e5·19-s − 2.37e4·20-s − 2.02e5·21-s + 4.41e5·22-s − 4.41e4·23-s + 8.16e5·24-s − 1.90e6·25-s + 3.04e5·26-s + 5.31e5·27-s − 2.66e5·28-s + ⋯ |
| L(s) = 1 | − 1.09·2-s + 0.577·3-s + 0.208·4-s − 0.158·5-s − 0.634·6-s − 0.392·7-s + 0.870·8-s + 0.333·9-s + 0.174·10-s − 0.365·11-s + 0.120·12-s − 0.118·13-s + 0.431·14-s − 0.0917·15-s − 1.16·16-s − 0.495·17-s − 0.366·18-s + 1.40·19-s − 0.0331·20-s − 0.226·21-s + 0.401·22-s − 0.0328·23-s + 0.502·24-s − 0.974·25-s + 0.130·26-s + 0.192·27-s − 0.0819·28-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
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 24.8T + 512T^{2} \) |
| 5 | \( 1 + 222.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 2.49e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.77e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.22e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.70e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.00e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 4.41e4T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.97e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.78e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.62e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 4.70e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.93e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.72e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 7.27e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 1.75e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.04e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 4.00e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.17e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.33e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.31e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 9.11e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 4.51e8T + 7.60e17T^{2} \) |
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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
−10.09428116919793837831042812997, −9.529224295218802845721851585024, −8.521109836101561487922125778076, −7.75086302099210217880679168661, −6.79509215539231640411855051222, −5.13483372446712665013345153188, −3.82034583886528960307533054547, −2.48555168464690775593154751169, −1.16943632805176769134895926694, 0,
1.16943632805176769134895926694, 2.48555168464690775593154751169, 3.82034583886528960307533054547, 5.13483372446712665013345153188, 6.79509215539231640411855051222, 7.75086302099210217880679168661, 8.521109836101561487922125778076, 9.529224295218802845721851585024, 10.09428116919793837831042812997
