| L(s) = 1 | − 2.13e4·2-s + 4.70e6·3-s + 3.19e8·4-s + 1.22e9·5-s − 1.00e11·6-s − 2.75e11·7-s − 3.95e12·8-s + 1.45e13·9-s − 2.60e13·10-s − 1.26e14·11-s + 1.50e15·12-s − 1.00e15·13-s + 5.86e15·14-s + 5.74e15·15-s + 4.13e16·16-s − 2.50e16·17-s − 3.10e17·18-s − 1.11e17·19-s + 3.90e17·20-s − 1.29e18·21-s + 2.69e18·22-s − 2.31e17·23-s − 1.86e19·24-s + 1.49e18·25-s + 2.14e19·26-s + 3.26e19·27-s − 8.80e19·28-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 1.70·3-s + 2.38·4-s + 0.447·5-s − 3.13·6-s − 1.07·7-s − 2.54·8-s + 1.90·9-s − 0.822·10-s − 1.10·11-s + 4.06·12-s − 0.923·13-s + 1.97·14-s + 0.762·15-s + 2.29·16-s − 0.614·17-s − 3.51·18-s − 0.609·19-s + 1.06·20-s − 1.83·21-s + 2.03·22-s − 0.0956·23-s − 4.34·24-s + 0.199·25-s + 1.69·26-s + 1.55·27-s − 2.55·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(14)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{29}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 1.22e9T \) |
| good | 2 | \( 1 + 2.13e4T + 1.34e8T^{2} \) |
| 3 | \( 1 - 4.70e6T + 7.62e12T^{2} \) |
| 7 | \( 1 + 2.75e11T + 6.57e22T^{2} \) |
| 11 | \( 1 + 1.26e14T + 1.31e28T^{2} \) |
| 13 | \( 1 + 1.00e15T + 1.19e30T^{2} \) |
| 17 | \( 1 + 2.50e16T + 1.66e33T^{2} \) |
| 19 | \( 1 + 1.11e17T + 3.36e34T^{2} \) |
| 23 | \( 1 + 2.31e17T + 5.84e36T^{2} \) |
| 29 | \( 1 - 3.28e19T + 3.05e39T^{2} \) |
| 31 | \( 1 + 1.48e20T + 1.84e40T^{2} \) |
| 37 | \( 1 - 2.40e20T + 2.19e42T^{2} \) |
| 41 | \( 1 - 2.11e21T + 3.50e43T^{2} \) |
| 43 | \( 1 + 1.09e22T + 1.26e44T^{2} \) |
| 47 | \( 1 + 7.41e22T + 1.40e45T^{2} \) |
| 53 | \( 1 - 2.60e23T + 3.59e46T^{2} \) |
| 59 | \( 1 - 7.05e23T + 6.50e47T^{2} \) |
| 61 | \( 1 + 1.00e23T + 1.59e48T^{2} \) |
| 67 | \( 1 + 7.88e24T + 2.01e49T^{2} \) |
| 71 | \( 1 - 1.67e25T + 9.63e49T^{2} \) |
| 73 | \( 1 + 5.00e24T + 2.04e50T^{2} \) |
| 79 | \( 1 - 2.49e25T + 1.72e51T^{2} \) |
| 83 | \( 1 + 1.43e26T + 6.53e51T^{2} \) |
| 89 | \( 1 - 2.45e26T + 4.30e52T^{2} \) |
| 97 | \( 1 + 2.77e26T + 4.39e53T^{2} \) |
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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
−16.30918458201456176066643376117, −15.05946671960574868215217446984, −13.04308088867297107835661044723, −10.23666517936782384566256991631, −9.388265511444021117592947741263, −8.225593358336636671553241578207, −6.93801418135339779160637816163, −2.90292323254855494762462876129, −2.02748686087092218438579694522, 0,
2.02748686087092218438579694522, 2.90292323254855494762462876129, 6.93801418135339779160637816163, 8.225593358336636671553241578207, 9.388265511444021117592947741263, 10.23666517936782384566256991631, 13.04308088867297107835661044723, 15.05946671960574868215217446984, 16.30918458201456176066643376117
