| L(s) = 1 | − 7-s − 3·9-s − 4·11-s + 3·13-s + 17-s − 23-s + 4·31-s − 11·37-s − 10·41-s − 2·43-s + 11·47-s + 49-s − 53-s + 8·59-s − 8·61-s + 3·63-s + 4·71-s + 4·73-s + 4·77-s + 11·79-s + 9·81-s − 13·83-s + 89-s − 3·91-s + 7·97-s + 12·99-s + 101-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s − 1.20·11-s + 0.832·13-s + 0.242·17-s − 0.208·23-s + 0.718·31-s − 1.80·37-s − 1.56·41-s − 0.304·43-s + 1.60·47-s + 1/7·49-s − 0.137·53-s + 1.04·59-s − 1.02·61-s + 0.377·63-s + 0.474·71-s + 0.468·73-s + 0.455·77-s + 1.23·79-s + 81-s − 1.42·83-s + 0.105·89-s − 0.314·91-s + 0.710·97-s + 1.20·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8754839944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8754839944\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.01584926357733, −13.69303829349894, −13.43206897715566, −12.71030831683902, −12.13527243748900, −11.85084723323661, −11.09955418434181, −10.66137975104343, −10.27578166419209, −9.722893662183073, −8.931935508840275, −8.618075494702589, −8.085084598764502, −7.628696894194893, −6.786268163803611, −6.464734572536015, −5.587212892580776, −5.443268141330724, −4.769764028919036, −3.855836724022520, −3.384812271331563, −2.776946919689350, −2.202235930902563, −1.300951037267081, −0.3184888509161935,
0.3184888509161935, 1.300951037267081, 2.202235930902563, 2.776946919689350, 3.384812271331563, 3.855836724022520, 4.769764028919036, 5.443268141330724, 5.587212892580776, 6.464734572536015, 6.786268163803611, 7.628696894194893, 8.085084598764502, 8.618075494702589, 8.931935508840275, 9.722893662183073, 10.27578166419209, 10.66137975104343, 11.09955418434181, 11.85084723323661, 12.13527243748900, 12.71030831683902, 13.43206897715566, 13.69303829349894, 14.01584926357733
