| L(s) = 1 | − 2·5-s + 7-s − 4·11-s − 13-s − 6·17-s + 4·19-s + 8·23-s − 25-s + 6·29-s + 8·31-s − 2·35-s + 2·37-s − 2·41-s − 4·43-s + 12·47-s + 49-s + 6·53-s + 8·55-s − 12·59-s − 6·61-s + 2·65-s − 16·71-s − 2·73-s − 4·77-s + 4·83-s + 12·85-s − 2·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 1.20·11-s − 0.277·13-s − 1.45·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.338·35-s + 0.328·37-s − 0.312·41-s − 0.609·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s + 1.07·55-s − 1.56·59-s − 0.768·61-s + 0.248·65-s − 1.89·71-s − 0.234·73-s − 0.455·77-s + 0.439·83-s + 1.30·85-s − 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.61138227150661299886474120026, −7.15058479585012122876172412971, −6.33746865755064908823612227695, −5.33311206879602121953924418003, −4.72639825567634000418862949259, −4.19237928482593613286833254108, −2.98810415480492252830043880919, −2.56814021450321899428515125912, −1.15840199401338845116090360872, 0,
1.15840199401338845116090360872, 2.56814021450321899428515125912, 2.98810415480492252830043880919, 4.19237928482593613286833254108, 4.72639825567634000418862949259, 5.33311206879602121953924418003, 6.33746865755064908823612227695, 7.15058479585012122876172412971, 7.61138227150661299886474120026
