| L(s) = 1 | − 5-s + 6·13-s + 2·17-s + 2·19-s + 6·23-s + 25-s − 6·29-s − 4·31-s + 8·37-s + 8·41-s + 43-s + 6·47-s − 7·49-s − 6·53-s − 4·59-s + 14·61-s − 6·65-s − 4·67-s − 8·71-s − 4·73-s + 12·79-s + 2·83-s − 2·85-s − 14·89-s − 2·95-s − 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.66·13-s + 0.485·17-s + 0.458·19-s + 1.25·23-s + 1/5·25-s − 1.11·29-s − 0.718·31-s + 1.31·37-s + 1.24·41-s + 0.152·43-s + 0.875·47-s − 49-s − 0.824·53-s − 0.520·59-s + 1.79·61-s − 0.744·65-s − 0.488·67-s − 0.949·71-s − 0.468·73-s + 1.35·79-s + 0.219·83-s − 0.216·85-s − 1.48·89-s − 0.205·95-s − 0.203·97-s + 0.0995·101-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 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.74072588650227, −13.13243065596134, −12.91944272984906, −12.46336569859247, −11.64270184115048, −11.35267929106969, −10.94418168402327, −10.60158625287427, −9.799290744381303, −9.264525837210279, −8.996779637661287, −8.346053022617049, −7.824244017359317, −7.447811265639239, −6.860421420861217, −6.223331285747393, −5.775163572175530, −5.287138574983603, −4.592048426936206, −3.939976843185767, −3.582437634624126, −2.980997849354215, −2.325780706628375, −1.264944675662694, −1.090170352971485, 0,
1.090170352971485, 1.264944675662694, 2.325780706628375, 2.980997849354215, 3.582437634624126, 3.939976843185767, 4.592048426936206, 5.287138574983603, 5.775163572175530, 6.223331285747393, 6.860421420861217, 7.447811265639239, 7.824244017359317, 8.346053022617049, 8.996779637661287, 9.264525837210279, 9.799290744381303, 10.60158625287427, 10.94418168402327, 11.35267929106969, 11.64270184115048, 12.46336569859247, 12.91944272984906, 13.13243065596134, 13.74072588650227
