| L(s) = 1 | − 3-s − 5-s − 7-s − 2·9-s + 11-s + 13-s + 15-s − 2·17-s − 4·19-s + 21-s + 6·23-s − 4·25-s + 5·27-s − 5·31-s − 33-s + 35-s + 4·37-s − 39-s + 4·41-s − 11·43-s + 2·45-s + 3·47-s + 49-s + 2·51-s + 3·53-s − 55-s + 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.301·11-s + 0.277·13-s + 0.258·15-s − 0.485·17-s − 0.917·19-s + 0.218·21-s + 1.25·23-s − 4/5·25-s + 0.962·27-s − 0.898·31-s − 0.174·33-s + 0.169·35-s + 0.657·37-s − 0.160·39-s + 0.624·41-s − 1.67·43-s + 0.298·45-s + 0.437·47-s + 1/7·49-s + 0.280·51-s + 0.412·53-s − 0.134·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6642670674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6642670674\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.81782999826392, −13.15352676397672, −12.87750255847678, −12.33476799032298, −11.73112304299220, −11.34027444307668, −11.01081375114588, −10.52134486538161, −9.867730586811117, −9.316310895527902, −8.721952899698795, −8.478687181790897, −7.746952872825839, −7.180069836502925, −6.562164567544453, −6.289016025023198, −5.587202215009904, −5.154208635294327, −4.447527318232177, −3.915695433816171, −3.339521125253487, −2.675081049981354, −2.009600446182235, −1.103934018297256, −0.2959672929474769,
0.2959672929474769, 1.103934018297256, 2.009600446182235, 2.675081049981354, 3.339521125253487, 3.915695433816171, 4.447527318232177, 5.154208635294327, 5.587202215009904, 6.289016025023198, 6.562164567544453, 7.180069836502925, 7.746952872825839, 8.478687181790897, 8.721952899698795, 9.316310895527902, 9.867730586811117, 10.52134486538161, 11.01081375114588, 11.34027444307668, 11.73112304299220, 12.33476799032298, 12.87750255847678, 13.15352676397672, 13.81782999826392
