| L(s) = 1 | − 5-s − 2·13-s + 17-s + 8·19-s + 9·23-s + 25-s + 2·29-s − 9·31-s − 8·37-s + 10·41-s − 6·43-s + 7·47-s − 7·49-s − 3·53-s + 6·59-s − 3·61-s + 2·65-s + 4·67-s + 12·71-s + 12·73-s + 11·79-s − 16·83-s − 85-s − 4·89-s − 8·95-s + 8·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.554·13-s + 0.242·17-s + 1.83·19-s + 1.87·23-s + 1/5·25-s + 0.371·29-s − 1.61·31-s − 1.31·37-s + 1.56·41-s − 0.914·43-s + 1.02·47-s − 49-s − 0.412·53-s + 0.781·59-s − 0.384·61-s + 0.248·65-s + 0.488·67-s + 1.42·71-s + 1.40·73-s + 1.23·79-s − 1.75·83-s − 0.108·85-s − 0.423·89-s − 0.820·95-s + 0.812·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.041666691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.041666691\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.54278916995061, −15.06318792821256, −14.38717259717593, −14.08823010517631, −13.36088546205957, −12.72079423140562, −12.34458861225087, −11.73229205451897, −11.07982574222496, −10.82693117900540, −9.911442522612483, −9.408761512121369, −9.000317422773701, −8.185599968164258, −7.608793445933057, −7.085788964584479, −6.690041278945815, −5.513196646580782, −5.275883592939425, −4.612109731002980, −3.623805476709409, −3.243043066025769, −2.445753390608944, −1.405750273501527, −0.6202062904258654,
0.6202062904258654, 1.405750273501527, 2.445753390608944, 3.243043066025769, 3.623805476709409, 4.612109731002980, 5.275883592939425, 5.513196646580782, 6.690041278945815, 7.085788964584479, 7.608793445933057, 8.185599968164258, 9.000317422773701, 9.408761512121369, 9.911442522612483, 10.82693117900540, 11.07982574222496, 11.73229205451897, 12.34458861225087, 12.72079423140562, 13.36088546205957, 14.08823010517631, 14.38717259717593, 15.06318792821256, 15.54278916995061
