| L(s) = 1 | − 8.68·3-s − 3.56·5-s + 27.1·7-s + 48.4·9-s − 15.2·11-s − 13·13-s + 30.9·15-s + 44.5·17-s − 23.9·19-s − 236.·21-s − 122.·23-s − 112.·25-s − 186.·27-s − 219.·29-s − 27.0·31-s + 132.·33-s − 96.7·35-s + 94.1·37-s + 112.·39-s − 160.·41-s + 151.·43-s − 172.·45-s − 466.·47-s + 395.·49-s − 386.·51-s − 120.·53-s + 54.3·55-s + ⋯ |
| L(s) = 1 | − 1.67·3-s − 0.318·5-s + 1.46·7-s + 1.79·9-s − 0.418·11-s − 0.277·13-s + 0.532·15-s + 0.635·17-s − 0.289·19-s − 2.45·21-s − 1.11·23-s − 0.898·25-s − 1.32·27-s − 1.40·29-s − 0.156·31-s + 0.699·33-s − 0.467·35-s + 0.418·37-s + 0.463·39-s − 0.610·41-s + 0.536·43-s − 0.571·45-s − 1.44·47-s + 1.15·49-s − 1.06·51-s − 0.313·53-s + 0.133·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 3 | \( 1 + 8.68T + 27T^{2} \) |
| 5 | \( 1 + 3.56T + 125T^{2} \) |
| 7 | \( 1 - 27.1T + 343T^{2} \) |
| 11 | \( 1 + 15.2T + 1.33e3T^{2} \) |
| 17 | \( 1 - 44.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 23.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 122.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 219.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 27.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 94.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 160.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 151.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 466.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 120.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 439.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 137.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 512.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 410.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 308.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 586.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.35e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 439.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.51e3T + 9.12e5T^{2} \) |
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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
−11.51533502890607168260277187719, −10.78170541356509805002664158193, −9.839457269766339393644111305126, −8.141650479274049139063362431739, −7.35811872376366126176835721091, −5.94209012268120497054491524259, −5.17398079438253441113167640479, −4.18380560692029457266100247885, −1.66305058468006764603969257999, 0,
1.66305058468006764603969257999, 4.18380560692029457266100247885, 5.17398079438253441113167640479, 5.94209012268120497054491524259, 7.35811872376366126176835721091, 8.141650479274049139063362431739, 9.839457269766339393644111305126, 10.78170541356509805002664158193, 11.51533502890607168260277187719
