| L(s) = 1 | − 8.19e3·2-s − 3.70e6·3-s + 6.71e7·4-s − 1.22e9·5-s + 3.03e10·6-s + 2.21e11·7-s − 5.49e11·8-s + 6.07e12·9-s + 1.00e13·10-s − 4.96e12·11-s − 2.48e14·12-s − 2.20e14·13-s − 1.81e15·14-s + 4.51e15·15-s + 4.50e15·16-s − 2.82e16·17-s − 4.97e16·18-s − 3.08e17·19-s − 8.19e16·20-s − 8.21e17·21-s + 4.06e16·22-s − 1.01e18·23-s + 2.03e18·24-s + 1.49e18·25-s + 1.81e18·26-s + 5.73e18·27-s + 1.48e19·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.34·3-s + 0.5·4-s − 0.447·5-s + 0.947·6-s + 0.865·7-s − 0.353·8-s + 0.796·9-s + 0.316·10-s − 0.0433·11-s − 0.670·12-s − 0.202·13-s − 0.612·14-s + 0.599·15-s + 0.250·16-s − 0.692·17-s − 0.563·18-s − 1.68·19-s − 0.223·20-s − 1.16·21-s + 0.0306·22-s − 0.420·23-s + 0.473·24-s + 0.199·25-s + 0.143·26-s + 0.272·27-s + 0.432·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(14)\) |
\(\approx\) |
\(0.4537352077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4537352077\) |
| \(L(\frac{29}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8.19e3T \) |
| 5 | \( 1 + 1.22e9T \) |
| good | 3 | \( 1 + 3.70e6T + 7.62e12T^{2} \) |
| 7 | \( 1 - 2.21e11T + 6.57e22T^{2} \) |
| 11 | \( 1 + 4.96e12T + 1.31e28T^{2} \) |
| 13 | \( 1 + 2.20e14T + 1.19e30T^{2} \) |
| 17 | \( 1 + 2.82e16T + 1.66e33T^{2} \) |
| 19 | \( 1 + 3.08e17T + 3.36e34T^{2} \) |
| 23 | \( 1 + 1.01e18T + 5.84e36T^{2} \) |
| 29 | \( 1 + 1.98e19T + 3.05e39T^{2} \) |
| 31 | \( 1 + 2.14e20T + 1.84e40T^{2} \) |
| 37 | \( 1 + 1.51e21T + 2.19e42T^{2} \) |
| 41 | \( 1 - 6.15e21T + 3.50e43T^{2} \) |
| 43 | \( 1 - 1.65e22T + 1.26e44T^{2} \) |
| 47 | \( 1 + 4.19e21T + 1.40e45T^{2} \) |
| 53 | \( 1 - 2.49e23T + 3.59e46T^{2} \) |
| 59 | \( 1 - 9.83e23T + 6.50e47T^{2} \) |
| 61 | \( 1 + 3.10e23T + 1.59e48T^{2} \) |
| 67 | \( 1 + 5.68e24T + 2.01e49T^{2} \) |
| 71 | \( 1 + 1.00e25T + 9.63e49T^{2} \) |
| 73 | \( 1 - 8.83e24T + 2.04e50T^{2} \) |
| 79 | \( 1 + 5.62e25T + 1.72e51T^{2} \) |
| 83 | \( 1 - 1.02e26T + 6.53e51T^{2} \) |
| 89 | \( 1 + 1.49e26T + 4.30e52T^{2} \) |
| 97 | \( 1 + 5.13e26T + 4.39e53T^{2} \) |
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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
−14.81769808379749654003886499049, −12.53522425623574718283737721364, −11.32151222188806323740947560010, −10.62803714391003157110538564708, −8.676409457306965029706148662096, −7.19106737276169071822550298660, −5.79433923766414706649305681775, −4.36014088307428706865221189980, −1.96434064848990649254962466703, −0.44741907088208802246144618443,
0.44741907088208802246144618443, 1.96434064848990649254962466703, 4.36014088307428706865221189980, 5.79433923766414706649305681775, 7.19106737276169071822550298660, 8.676409457306965029706148662096, 10.62803714391003157110538564708, 11.32151222188806323740947560010, 12.53522425623574718283737721364, 14.81769808379749654003886499049
