| L(s) = 1 | + 2·3-s − 5-s + 9-s − 2·11-s + 2·13-s − 2·15-s + 4·17-s + 6·19-s + 8·23-s + 25-s − 4·27-s + 10·31-s − 4·33-s + 8·37-s + 4·39-s + 2·43-s − 45-s − 6·47-s − 7·49-s + 8·51-s + 6·53-s + 2·55-s + 12·57-s + 4·59-s − 12·61-s − 2·65-s − 4·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s + 0.554·13-s − 0.516·15-s + 0.970·17-s + 1.37·19-s + 1.66·23-s + 1/5·25-s − 0.769·27-s + 1.79·31-s − 0.696·33-s + 1.31·37-s + 0.640·39-s + 0.304·43-s − 0.149·45-s − 0.875·47-s − 49-s + 1.12·51-s + 0.824·53-s + 0.269·55-s + 1.58·57-s + 0.520·59-s − 1.53·61-s − 0.248·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.009182558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.009182558\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−15.01452303875633, −14.50198920056718, −13.93649449995628, −13.41652932242239, −13.18450854370768, −12.34400773316486, −11.89267967078454, −11.24113135078031, −10.81344991981626, −10.04145442651034, −9.466569369208585, −9.179179671055032, −8.343042792855262, −7.904422035169885, −7.739102820780444, −6.873517497638462, −6.288383767542329, −5.368279635390929, −5.010157895157111, −4.128282726361447, −3.434454759636040, −2.950736974384013, −2.574203108534329, −1.377146696436343, −0.7674892640674869,
0.7674892640674869, 1.377146696436343, 2.574203108534329, 2.950736974384013, 3.434454759636040, 4.128282726361447, 5.010157895157111, 5.368279635390929, 6.288383767542329, 6.873517497638462, 7.739102820780444, 7.904422035169885, 8.343042792855262, 9.179179671055032, 9.466569369208585, 10.04145442651034, 10.81344991981626, 11.24113135078031, 11.89267967078454, 12.34400773316486, 13.18450854370768, 13.41652932242239, 13.93649449995628, 14.50198920056718, 15.01452303875633
