| L(s) = 1 | − 3·5-s − 7-s − 2·11-s − 4·13-s + 5·17-s + 4·23-s + 4·25-s + 4·29-s + 8·31-s + 3·35-s + 37-s + 7·41-s − 5·43-s − 47-s + 49-s + 2·53-s + 6·55-s + 11·59-s − 14·61-s + 12·65-s + 4·67-s − 12·71-s + 2·77-s − 13·79-s − 11·83-s − 15·85-s + 6·89-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s − 0.603·11-s − 1.10·13-s + 1.21·17-s + 0.834·23-s + 4/5·25-s + 0.742·29-s + 1.43·31-s + 0.507·35-s + 0.164·37-s + 1.09·41-s − 0.762·43-s − 0.145·47-s + 1/7·49-s + 0.274·53-s + 0.809·55-s + 1.43·59-s − 1.79·61-s + 1.48·65-s + 0.488·67-s − 1.42·71-s + 0.227·77-s − 1.46·79-s − 1.20·83-s − 1.62·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.65657769928688365304753972703, −7.25397291424937042517392669427, −6.41456568925980098296449714335, −5.45574813709806874328256013639, −4.74539524833964419774676384523, −4.09541803336478800287702544014, −3.10391740876158810694687813712, −2.67418383327204401633220378167, −1.06644111108047369711507869826, 0,
1.06644111108047369711507869826, 2.67418383327204401633220378167, 3.10391740876158810694687813712, 4.09541803336478800287702544014, 4.74539524833964419774676384523, 5.45574813709806874328256013639, 6.41456568925980098296449714335, 7.25397291424937042517392669427, 7.65657769928688365304753972703
