| L(s) = 1 | + 2·3-s + 9-s − 4·11-s − 2·13-s − 2·19-s − 4·23-s − 4·27-s + 10·29-s − 4·31-s − 8·33-s + 2·37-s − 4·39-s − 12·41-s − 4·43-s + 4·47-s − 2·53-s − 4·57-s − 10·59-s + 6·61-s + 4·67-s − 8·69-s + 12·71-s + 4·73-s + 4·79-s − 11·81-s + 14·83-s + 20·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.458·19-s − 0.834·23-s − 0.769·27-s + 1.85·29-s − 0.718·31-s − 1.39·33-s + 0.328·37-s − 0.640·39-s − 1.87·41-s − 0.609·43-s + 0.583·47-s − 0.274·53-s − 0.529·57-s − 1.30·59-s + 0.768·61-s + 0.488·67-s − 0.963·69-s + 1.42·71-s + 0.468·73-s + 0.450·79-s − 1.22·81-s + 1.53·83-s + 2.14·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.216287585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.216287585\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−15.54929936500074, −15.20432563330448, −14.47828766739144, −14.13406910919750, −13.53574472676541, −13.15474309099334, −12.42227901732345, −12.00926891107638, −11.22676898978041, −10.50538101642476, −10.08686844899767, −9.558993134938923, −8.792817925290332, −8.327827629190054, −7.914082376484230, −7.345479860680235, −6.576633048393419, −5.912355053982715, −5.016739468055870, −4.651522777989891, −3.573678612971159, −3.184019069941458, −2.280209565528670, −2.006534568523315, −0.5401618805520083,
0.5401618805520083, 2.006534568523315, 2.280209565528670, 3.184019069941458, 3.573678612971159, 4.651522777989891, 5.016739468055870, 5.912355053982715, 6.576633048393419, 7.345479860680235, 7.914082376484230, 8.327827629190054, 8.792817925290332, 9.558993134938923, 10.08686844899767, 10.50538101642476, 11.22676898978041, 12.00926891107638, 12.42227901732345, 13.15474309099334, 13.53574472676541, 14.13406910919750, 14.47828766739144, 15.20432563330448, 15.54929936500074
