| L(s) = 1 | − 5-s − 2·7-s + 6·11-s − 2·13-s + 4·17-s − 6·19-s + 6·23-s + 25-s + 10·29-s + 2·35-s − 2·37-s + 6·41-s − 43-s + 6·47-s − 3·49-s + 2·53-s − 6·55-s − 14·59-s + 8·61-s + 2·65-s − 4·67-s − 12·73-s − 12·77-s − 4·83-s − 4·85-s + 10·89-s + 4·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 1.80·11-s − 0.554·13-s + 0.970·17-s − 1.37·19-s + 1.25·23-s + 1/5·25-s + 1.85·29-s + 0.338·35-s − 0.328·37-s + 0.937·41-s − 0.152·43-s + 0.875·47-s − 3/7·49-s + 0.274·53-s − 0.809·55-s − 1.82·59-s + 1.02·61-s + 0.248·65-s − 0.488·67-s − 1.40·73-s − 1.36·77-s − 0.439·83-s − 0.433·85-s + 1.05·89-s + 0.419·91-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 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.94061824989619, −13.17781954846821, −12.69681059094297, −12.30144862067266, −11.94732840396867, −11.50156075085087, −10.79067159452029, −10.43527660837803, −9.848362370539752, −9.350407251752418, −8.876471108412412, −8.552879946026467, −7.821130690050638, −7.261439559944314, −6.742060785555522, −6.402815244559510, −5.936561011470316, −5.128975075961331, −4.472523649813684, −4.157151570275581, −3.451954324846227, −2.993063366381634, −2.368096441291533, −1.370152658663828, −0.9359741619059174, 0,
0.9359741619059174, 1.370152658663828, 2.368096441291533, 2.993063366381634, 3.451954324846227, 4.157151570275581, 4.472523649813684, 5.128975075961331, 5.936561011470316, 6.402815244559510, 6.742060785555522, 7.261439559944314, 7.821130690050638, 8.552879946026467, 8.876471108412412, 9.350407251752418, 9.848362370539752, 10.43527660837803, 10.79067159452029, 11.50156075085087, 11.94732840396867, 12.30144862067266, 12.69681059094297, 13.17781954846821, 13.94061824989619
