| L(s) = 1 | + 2·5-s − 3·7-s + 2·11-s + 13-s − 8·17-s − 7·19-s − 4·23-s − 25-s + 6·29-s − 31-s − 6·35-s + 3·37-s − 6·41-s + 43-s − 6·47-s + 2·49-s + 2·53-s + 4·55-s − 10·59-s + 10·61-s + 2·65-s + 8·67-s − 14·71-s + 10·73-s − 6·77-s − 11·79-s + 4·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.13·7-s + 0.603·11-s + 0.277·13-s − 1.94·17-s − 1.60·19-s − 0.834·23-s − 1/5·25-s + 1.11·29-s − 0.179·31-s − 1.01·35-s + 0.493·37-s − 0.937·41-s + 0.152·43-s − 0.875·47-s + 2/7·49-s + 0.274·53-s + 0.539·55-s − 1.30·59-s + 1.28·61-s + 0.248·65-s + 0.977·67-s − 1.66·71-s + 1.17·73-s − 0.683·77-s − 1.23·79-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−8.868077231526493431741667108389, −8.268283177423319799702359334990, −6.77741791228600683337894420650, −6.49976612642460087907127194344, −5.88478303359720193369907217614, −4.59646513756340349672120012748, −3.86012827248629715550125307113, −2.63542411493964303752276019082, −1.81068575808146793096582213511, 0,
1.81068575808146793096582213511, 2.63542411493964303752276019082, 3.86012827248629715550125307113, 4.59646513756340349672120012748, 5.88478303359720193369907217614, 6.49976612642460087907127194344, 6.77741791228600683337894420650, 8.268283177423319799702359334990, 8.868077231526493431741667108389
