| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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
−14.01108773988815, −13.19290631635547, −12.86627771999658, −12.24615716391780, −11.96095777955131, −11.31248679687865, −10.86492666383765, −10.42939770720198, −9.893418544070806, −9.360488246308977, −8.970048112369433, −8.269920489583476, −7.927475941858863, −7.505628719306026, −6.583155274752400, −6.331045213176251, −5.900250809256650, −5.120159777609618, −4.601961480613524, −4.162370577836867, −3.656188870626864, −2.639752165267518, −2.209541730136460, −1.723555624566136, −0.9104042009626251, 0,
0.9104042009626251, 1.723555624566136, 2.209541730136460, 2.639752165267518, 3.656188870626864, 4.162370577836867, 4.601961480613524, 5.120159777609618, 5.900250809256650, 6.331045213176251, 6.583155274752400, 7.505628719306026, 7.927475941858863, 8.269920489583476, 8.970048112369433, 9.360488246308977, 9.893418544070806, 10.42939770720198, 10.86492666383765, 11.31248679687865, 11.96095777955131, 12.24615716391780, 12.86627771999658, 13.19290631635547, 14.01108773988815
