| L(s) = 1 | − 3-s + 9-s + 11-s + 4·13-s + 3·17-s + 3·19-s − 3·23-s − 27-s + 9·29-s + 6·31-s − 33-s + 8·37-s − 4·39-s + 4·41-s + 43-s − 6·47-s − 3·51-s − 11·53-s − 3·57-s − 3·59-s + 61-s + 2·67-s + 3·69-s + 6·71-s + 14·73-s − 4·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 0.727·17-s + 0.688·19-s − 0.625·23-s − 0.192·27-s + 1.67·29-s + 1.07·31-s − 0.174·33-s + 1.31·37-s − 0.640·39-s + 0.624·41-s + 0.152·43-s − 0.875·47-s − 0.420·51-s − 1.51·53-s − 0.397·57-s − 0.390·59-s + 0.128·61-s + 0.244·67-s + 0.361·69-s + 0.712·71-s + 1.63·73-s − 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.465856593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.465856593\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−12.59653452908973, −12.06927691221401, −11.76526725848812, −11.17888637238750, −10.98967339003590, −10.29613535567869, −9.909659964666732, −9.552058872823240, −9.017172775210520, −8.331600839919674, −7.988136238067874, −7.704952571093991, −6.767802207081525, −6.598093309978527, −5.984222937055674, −5.762801277524384, −5.002013871352990, −4.553364917645106, −4.141235090499726, −3.361359265328733, −3.104797517116459, −2.313843424021094, −1.578597966706942, −0.9586877532801911, −0.6425208973549215,
0.6425208973549215, 0.9586877532801911, 1.578597966706942, 2.313843424021094, 3.104797517116459, 3.361359265328733, 4.141235090499726, 4.553364917645106, 5.002013871352990, 5.762801277524384, 5.984222937055674, 6.598093309978527, 6.767802207081525, 7.704952571093991, 7.988136238067874, 8.331600839919674, 9.017172775210520, 9.552058872823240, 9.909659964666732, 10.29613535567869, 10.98967339003590, 11.17888637238750, 11.76526725848812, 12.06927691221401, 12.59653452908973
