| L(s) = 1 | − 5-s − 2·11-s − 3·17-s − 19-s − 3·23-s + 25-s − 4·29-s − 5·31-s + 10·37-s − 6·41-s − 6·43-s − 8·47-s − 7·49-s − 3·53-s + 2·55-s + 5·61-s − 2·67-s − 2·71-s + 6·73-s − 11·79-s − 9·83-s + 3·85-s − 10·89-s + 95-s + 8·97-s + 12·101-s − 12·103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.603·11-s − 0.727·17-s − 0.229·19-s − 0.625·23-s + 1/5·25-s − 0.742·29-s − 0.898·31-s + 1.64·37-s − 0.937·41-s − 0.914·43-s − 1.16·47-s − 49-s − 0.412·53-s + 0.269·55-s + 0.640·61-s − 0.244·67-s − 0.237·71-s + 0.702·73-s − 1.23·79-s − 0.987·83-s + 0.325·85-s − 1.05·89-s + 0.102·95-s + 0.812·97-s + 1.19·101-s − 1.18·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−9.513431316514600408003072092907, −8.523146712604219482786278195739, −7.88476640049411651853631457862, −7.00481999528092178116170950569, −6.09524055825850124940401922312, −5.07264494567311717754515914838, −4.17466568064428202291161107248, −3.12527269276681833472760842290, −1.88724812338382557411512712411, 0,
1.88724812338382557411512712411, 3.12527269276681833472760842290, 4.17466568064428202291161107248, 5.07264494567311717754515914838, 6.09524055825850124940401922312, 7.00481999528092178116170950569, 7.88476640049411651853631457862, 8.523146712604219482786278195739, 9.513431316514600408003072092907
