| L(s) = 1 | + 4·5-s + 3·7-s − 4·11-s + 13-s − 4·17-s + 19-s − 4·23-s + 11·25-s + 4·31-s + 12·35-s − 9·37-s + 8·43-s + 12·47-s + 2·49-s − 8·53-s − 16·55-s − 4·59-s − 5·61-s + 4·65-s − 11·67-s − 8·71-s + 73-s − 12·77-s + 5·79-s − 8·83-s − 16·85-s + 12·89-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1.13·7-s − 1.20·11-s + 0.277·13-s − 0.970·17-s + 0.229·19-s − 0.834·23-s + 11/5·25-s + 0.718·31-s + 2.02·35-s − 1.47·37-s + 1.21·43-s + 1.75·47-s + 2/7·49-s − 1.09·53-s − 2.15·55-s − 0.520·59-s − 0.640·61-s + 0.496·65-s − 1.34·67-s − 0.949·71-s + 0.117·73-s − 1.36·77-s + 0.562·79-s − 0.878·83-s − 1.73·85-s + 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.909052801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.909052801\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−10.77240220322663913758630781467, −10.46100545932507204746750392342, −9.353481303159343167683292410657, −8.539782519934566925988068508340, −7.50312413977084454112831481379, −6.23424820798207238945321961084, −5.45381403020204823365032144634, −4.60013470795046618235360781298, −2.61810189422413157391267840739, −1.69047687282002906512933211555,
1.69047687282002906512933211555, 2.61810189422413157391267840739, 4.60013470795046618235360781298, 5.45381403020204823365032144634, 6.23424820798207238945321961084, 7.50312413977084454112831481379, 8.539782519934566925988068508340, 9.353481303159343167683292410657, 10.46100545932507204746750392342, 10.77240220322663913758630781467
