| L(s) = 1 | − 9.99e3·2-s + 1.59e6·3-s − 3.42e7·4-s − 1.80e8·5-s − 1.59e10·6-s + 1.89e11·7-s + 1.68e12·8-s + 2.54e12·9-s + 1.80e12·10-s − 1.01e14·11-s − 5.46e13·12-s − 8.69e14·13-s − 1.89e15·14-s − 2.87e14·15-s − 1.22e16·16-s − 3.74e16·17-s − 2.54e16·18-s − 3.18e17·19-s + 6.17e15·20-s + 3.01e17·21-s + 1.01e18·22-s + 2.65e18·23-s + 2.68e18·24-s − 7.41e18·25-s + 8.69e18·26-s + 4.05e18·27-s − 6.48e18·28-s + ⋯ |
| L(s) = 1 | − 0.862·2-s + 0.577·3-s − 0.255·4-s − 0.0660·5-s − 0.498·6-s + 0.737·7-s + 1.08·8-s + 0.333·9-s + 0.0569·10-s − 0.884·11-s − 0.147·12-s − 0.796·13-s − 0.636·14-s − 0.0381·15-s − 0.679·16-s − 0.916·17-s − 0.287·18-s − 1.73·19-s + 0.0168·20-s + 0.426·21-s + 0.763·22-s + 1.09·23-s + 0.625·24-s − 0.995·25-s + 0.687·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 1.59e6T \) |
| good | 2 | \( 1 + 9.99e3T + 1.34e8T^{2} \) |
| 5 | \( 1 + 1.80e8T + 7.45e18T^{2} \) |
| 7 | \( 1 - 1.89e11T + 6.57e22T^{2} \) |
| 11 | \( 1 + 1.01e14T + 1.31e28T^{2} \) |
| 13 | \( 1 + 8.69e14T + 1.19e30T^{2} \) |
| 17 | \( 1 + 3.74e16T + 1.66e33T^{2} \) |
| 19 | \( 1 + 3.18e17T + 3.36e34T^{2} \) |
| 23 | \( 1 - 2.65e18T + 5.84e36T^{2} \) |
| 29 | \( 1 + 5.10e18T + 3.05e39T^{2} \) |
| 31 | \( 1 + 1.58e20T + 1.84e40T^{2} \) |
| 37 | \( 1 - 1.42e21T + 2.19e42T^{2} \) |
| 41 | \( 1 + 6.09e20T + 3.50e43T^{2} \) |
| 43 | \( 1 + 1.63e22T + 1.26e44T^{2} \) |
| 47 | \( 1 - 6.66e22T + 1.40e45T^{2} \) |
| 53 | \( 1 - 1.93e23T + 3.59e46T^{2} \) |
| 59 | \( 1 + 1.21e24T + 6.50e47T^{2} \) |
| 61 | \( 1 + 1.85e24T + 1.59e48T^{2} \) |
| 67 | \( 1 - 1.28e23T + 2.01e49T^{2} \) |
| 71 | \( 1 + 1.63e25T + 9.63e49T^{2} \) |
| 73 | \( 1 - 3.17e24T + 2.04e50T^{2} \) |
| 79 | \( 1 - 2.42e25T + 1.72e51T^{2} \) |
| 83 | \( 1 - 2.04e24T + 6.53e51T^{2} \) |
| 89 | \( 1 - 1.76e26T + 4.30e52T^{2} \) |
| 97 | \( 1 - 9.85e26T + 4.39e53T^{2} \) |
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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
−18.48578427542926461750281328292, −17.12264849363198900809883350514, −14.99663130486949289666488887069, −13.21909217079641720774820870763, −10.64752720863661838682538894663, −8.929021207689181822044436972339, −7.63813961817774875535686408725, −4.58973508480773331284927121023, −2.03829725232267675684035513441, 0,
2.03829725232267675684035513441, 4.58973508480773331284927121023, 7.63813961817774875535686408725, 8.929021207689181822044436972339, 10.64752720863661838682538894663, 13.21909217079641720774820870763, 14.99663130486949289666488887069, 17.12264849363198900809883350514, 18.48578427542926461750281328292
