| L(s) = 1 | − 5-s − 7-s + 3·11-s + 4·13-s + 6·17-s + 19-s + 3·23-s + 25-s + 31-s + 35-s + 4·37-s + 43-s + 12·47-s − 6·49-s + 9·53-s − 3·55-s + 10·61-s − 4·65-s + 10·67-s − 15·71-s − 7·73-s − 3·77-s + 5·79-s − 6·83-s − 6·85-s − 15·89-s − 4·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.904·11-s + 1.10·13-s + 1.45·17-s + 0.229·19-s + 0.625·23-s + 1/5·25-s + 0.179·31-s + 0.169·35-s + 0.657·37-s + 0.152·43-s + 1.75·47-s − 6/7·49-s + 1.23·53-s − 0.404·55-s + 1.28·61-s − 0.496·65-s + 1.22·67-s − 1.78·71-s − 0.819·73-s − 0.341·77-s + 0.562·79-s − 0.658·83-s − 0.650·85-s − 1.58·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.355742570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.355742570\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−13.94590847913856, −13.40036984405540, −12.79268645433368, −12.44572328460261, −11.88331658566213, −11.33561381861593, −11.15046333999812, −10.23791359107491, −10.02221450945494, −9.342612904917263, −8.809075767937565, −8.445879522061161, −7.814217530113987, −7.198246787074151, −6.873075940384310, −6.082169052297953, −5.759590638744122, −5.167712475028388, −4.213647150302554, −4.007818933337258, −3.246721276249019, −2.924381850668159, −1.892320462107714, −1.071077285250344, −0.7219853824307759,
0.7219853824307759, 1.071077285250344, 1.892320462107714, 2.924381850668159, 3.246721276249019, 4.007818933337258, 4.213647150302554, 5.167712475028388, 5.759590638744122, 6.082169052297953, 6.873075940384310, 7.198246787074151, 7.814217530113987, 8.445879522061161, 8.809075767937565, 9.342612904917263, 10.02221450945494, 10.23791359107491, 11.15046333999812, 11.33561381861593, 11.88331658566213, 12.44572328460261, 12.79268645433368, 13.40036984405540, 13.94590847913856
