| L(s) = 1 | − 2·4-s − 7-s − 6·11-s + 4·13-s + 4·16-s + 3·17-s + 2·19-s − 6·23-s + 2·28-s + 6·29-s − 4·31-s + 7·37-s + 3·41-s + 43-s + 12·44-s + 9·47-s + 49-s − 8·52-s − 6·53-s − 9·59-s − 10·61-s − 8·64-s + 4·67-s − 6·68-s − 2·73-s − 4·76-s + 6·77-s + ⋯ |
| L(s) = 1 | − 4-s − 0.377·7-s − 1.80·11-s + 1.10·13-s + 16-s + 0.727·17-s + 0.458·19-s − 1.25·23-s + 0.377·28-s + 1.11·29-s − 0.718·31-s + 1.15·37-s + 0.468·41-s + 0.152·43-s + 1.80·44-s + 1.31·47-s + 1/7·49-s − 1.10·52-s − 0.824·53-s − 1.17·59-s − 1.28·61-s − 64-s + 0.488·67-s − 0.727·68-s − 0.234·73-s − 0.458·76-s + 0.683·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.84992044053964800085190220961, −7.61770564757961554271645892692, −6.19833528683953656752348977000, −5.74521170629862645004981667515, −5.00255743126005056194877093074, −4.18667550940177211931958995326, −3.37523251394481359790706196138, −2.59135104534746647872303823771, −1.15394968040695473674965847739, 0,
1.15394968040695473674965847739, 2.59135104534746647872303823771, 3.37523251394481359790706196138, 4.18667550940177211931958995326, 5.00255743126005056194877093074, 5.74521170629862645004981667515, 6.19833528683953656752348977000, 7.61770564757961554271645892692, 7.84992044053964800085190220961
