| L(s) = 1 | − 5-s + 2·7-s − 2·11-s + 2·13-s + 8·19-s − 8·23-s + 25-s − 2·29-s − 2·35-s − 8·37-s + 8·41-s + 43-s − 3·49-s − 2·53-s + 2·55-s + 6·59-s + 8·61-s − 2·65-s − 12·67-s − 2·73-s − 4·77-s + 12·79-s − 18·83-s − 6·89-s + 4·91-s − 8·95-s − 2·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s − 0.603·11-s + 0.554·13-s + 1.83·19-s − 1.66·23-s + 1/5·25-s − 0.371·29-s − 0.338·35-s − 1.31·37-s + 1.24·41-s + 0.152·43-s − 3/7·49-s − 0.274·53-s + 0.269·55-s + 0.781·59-s + 1.02·61-s − 0.248·65-s − 1.46·67-s − 0.234·73-s − 0.455·77-s + 1.35·79-s − 1.97·83-s − 0.635·89-s + 0.419·91-s − 0.820·95-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.59021814025869, −14.71412387677861, −14.30631433479854, −13.84569777000078, −13.34131239069849, −12.67537368968635, −12.03348279632258, −11.68499330691281, −11.15794481294379, −10.62175960067797, −9.988708530563534, −9.505318002102385, −8.731813473304244, −8.233460897561101, −7.682700312507934, −7.366206502942887, −6.556239727711692, −5.699647093566488, −5.413335654663734, −4.658213156648561, −3.982837057338385, −3.413361110882083, −2.641815405691415, −1.792379043987789, −1.073338809457541, 0,
1.073338809457541, 1.792379043987789, 2.641815405691415, 3.413361110882083, 3.982837057338385, 4.658213156648561, 5.413335654663734, 5.699647093566488, 6.556239727711692, 7.366206502942887, 7.682700312507934, 8.233460897561101, 8.731813473304244, 9.505318002102385, 9.988708530563534, 10.62175960067797, 11.15794481294379, 11.68499330691281, 12.03348279632258, 12.67537368968635, 13.34131239069849, 13.84569777000078, 14.30631433479854, 14.71412387677861, 15.59021814025869
