| L(s) = 1 | − 7.95e3·2-s + 8.29e5·3-s − 7.08e7·4-s + 1.22e9·5-s − 6.60e9·6-s − 2.73e10·7-s + 1.63e12·8-s − 6.93e12·9-s − 9.71e12·10-s + 6.42e13·11-s − 5.88e13·12-s + 1.71e15·13-s + 2.17e14·14-s + 1.01e15·15-s − 3.48e15·16-s + 1.17e15·17-s + 5.52e16·18-s − 2.79e17·19-s − 8.64e16·20-s − 2.27e16·21-s − 5.11e17·22-s − 3.09e18·23-s + 1.35e18·24-s + 1.49e18·25-s − 1.36e19·26-s − 1.20e19·27-s + 1.94e18·28-s + ⋯ |
| L(s) = 1 | − 0.687·2-s + 0.300·3-s − 0.527·4-s + 0.447·5-s − 0.206·6-s − 0.106·7-s + 1.04·8-s − 0.909·9-s − 0.307·10-s + 0.561·11-s − 0.158·12-s + 1.56·13-s + 0.0733·14-s + 0.134·15-s − 0.193·16-s + 0.0287·17-s + 0.625·18-s − 1.52·19-s − 0.236·20-s − 0.0320·21-s − 0.385·22-s − 1.27·23-s + 0.315·24-s + 0.199·25-s − 1.07·26-s − 0.573·27-s + 0.0563·28-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 | \( 1 + 7.95e3T + 1.34e8T^{2} \) |
| 3 | \( 1 - 8.29e5T + 7.62e12T^{2} \) |
| 7 | \( 1 + 2.73e10T + 6.57e22T^{2} \) |
| 11 | \( 1 - 6.42e13T + 1.31e28T^{2} \) |
| 13 | \( 1 - 1.71e15T + 1.19e30T^{2} \) |
| 17 | \( 1 - 1.17e15T + 1.66e33T^{2} \) |
| 19 | \( 1 + 2.79e17T + 3.36e34T^{2} \) |
| 23 | \( 1 + 3.09e18T + 5.84e36T^{2} \) |
| 29 | \( 1 + 7.42e19T + 3.05e39T^{2} \) |
| 31 | \( 1 + 1.27e20T + 1.84e40T^{2} \) |
| 37 | \( 1 - 8.47e20T + 2.19e42T^{2} \) |
| 41 | \( 1 + 1.17e21T + 3.50e43T^{2} \) |
| 43 | \( 1 - 3.56e21T + 1.26e44T^{2} \) |
| 47 | \( 1 - 9.14e21T + 1.40e45T^{2} \) |
| 53 | \( 1 + 2.22e23T + 3.59e46T^{2} \) |
| 59 | \( 1 + 6.15e23T + 6.50e47T^{2} \) |
| 61 | \( 1 - 1.92e24T + 1.59e48T^{2} \) |
| 67 | \( 1 + 5.40e24T + 2.01e49T^{2} \) |
| 71 | \( 1 - 1.94e24T + 9.63e49T^{2} \) |
| 73 | \( 1 + 1.12e25T + 2.04e50T^{2} \) |
| 79 | \( 1 + 1.22e25T + 1.72e51T^{2} \) |
| 83 | \( 1 - 4.79e25T + 6.53e51T^{2} \) |
| 89 | \( 1 - 1.66e26T + 4.30e52T^{2} \) |
| 97 | \( 1 + 5.46e26T + 4.39e53T^{2} \) |
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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
−16.58694773233183728397527642436, −14.44105540189221087103734838844, −13.21180679057991594308534819744, −10.93234987010626124115522750856, −9.266651108792605807378440153570, −8.264403747745638834962669940015, −6.00472494448965829369289926727, −3.87007075371049794182385065458, −1.71380610193653052328344350711, 0,
1.71380610193653052328344350711, 3.87007075371049794182385065458, 6.00472494448965829369289926727, 8.264403747745638834962669940015, 9.266651108792605807378440153570, 10.93234987010626124115522750856, 13.21180679057991594308534819744, 14.44105540189221087103734838844, 16.58694773233183728397527642436
