| L(s) = 1 | + 3·3-s + 4·5-s − 26·7-s + 9·9-s − 11·11-s + 32·13-s + 12·15-s + 74·17-s + 60·19-s − 78·21-s − 182·23-s − 109·25-s + 27·27-s + 90·29-s − 8·31-s − 33·33-s − 104·35-s + 66·37-s + 96·39-s + 422·41-s − 408·43-s + 36·45-s − 506·47-s + 333·49-s + 222·51-s − 348·53-s − 44·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.357·5-s − 1.40·7-s + 1/3·9-s − 0.301·11-s + 0.682·13-s + 0.206·15-s + 1.05·17-s + 0.724·19-s − 0.810·21-s − 1.64·23-s − 0.871·25-s + 0.192·27-s + 0.576·29-s − 0.0463·31-s − 0.174·33-s − 0.502·35-s + 0.293·37-s + 0.394·39-s + 1.60·41-s − 1.44·43-s + 0.119·45-s − 1.57·47-s + 0.970·49-s + 0.609·51-s − 0.901·53-s − 0.107·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 11 | \( 1 + p T \) |
| good | 5 | \( 1 - 4 T + p^{3} T^{2} \) |
| 7 | \( 1 + 26 T + p^{3} T^{2} \) |
| 13 | \( 1 - 32 T + p^{3} T^{2} \) |
| 17 | \( 1 - 74 T + p^{3} T^{2} \) |
| 19 | \( 1 - 60 T + p^{3} T^{2} \) |
| 23 | \( 1 + 182 T + p^{3} T^{2} \) |
| 29 | \( 1 - 90 T + p^{3} T^{2} \) |
| 31 | \( 1 + 8 T + p^{3} T^{2} \) |
| 37 | \( 1 - 66 T + p^{3} T^{2} \) |
| 41 | \( 1 - 422 T + p^{3} T^{2} \) |
| 43 | \( 1 + 408 T + p^{3} T^{2} \) |
| 47 | \( 1 + 506 T + p^{3} T^{2} \) |
| 53 | \( 1 + 348 T + p^{3} T^{2} \) |
| 59 | \( 1 - 200 T + p^{3} T^{2} \) |
| 61 | \( 1 + 132 T + p^{3} T^{2} \) |
| 67 | \( 1 - 1036 T + p^{3} T^{2} \) |
| 71 | \( 1 - 762 T + p^{3} T^{2} \) |
| 73 | \( 1 + 542 T + p^{3} T^{2} \) |
| 79 | \( 1 + 550 T + p^{3} T^{2} \) |
| 83 | \( 1 - 132 T + p^{3} T^{2} \) |
| 89 | \( 1 - 570 T + p^{3} T^{2} \) |
| 97 | \( 1 - 14 T + p^{3} T^{2} \) |
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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
−8.213967367680691300782723569916, −7.79449479861081698286700699396, −6.66022364838920303122687000699, −6.10248769674964699522562224397, −5.29682195345652693109702144393, −3.94927740677310572827003028734, −3.36199407088840525199809496499, −2.50193778729963704125352144638, −1.32505436075178096330294889103, 0,
1.32505436075178096330294889103, 2.50193778729963704125352144638, 3.36199407088840525199809496499, 3.94927740677310572827003028734, 5.29682195345652693109702144393, 6.10248769674964699522562224397, 6.66022364838920303122687000699, 7.79449479861081698286700699396, 8.213967367680691300782723569916
