| L(s) = 1 | + 3-s − 2·5-s + 7-s + 9-s − 2·11-s − 2·15-s − 4·17-s − 4·19-s + 21-s + 6·23-s − 25-s + 27-s − 6·29-s − 31-s − 2·33-s − 2·35-s − 10·37-s − 4·41-s − 43-s − 2·45-s − 10·47-s − 6·49-s − 4·51-s − 8·53-s + 4·55-s − 4·57-s + 2·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s − 0.516·15-s − 0.970·17-s − 0.917·19-s + 0.218·21-s + 1.25·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.179·31-s − 0.348·33-s − 0.338·35-s − 1.64·37-s − 0.624·41-s − 0.152·43-s − 0.298·45-s − 1.45·47-s − 6/7·49-s − 0.560·51-s − 1.09·53-s + 0.539·55-s − 0.529·57-s + 0.260·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.056662052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.056662052\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−15.24295049332907, −14.56968845228492, −14.07972480051335, −13.38816891342007, −12.84735990557302, −12.67220187064692, −11.74584765820728, −11.21200878197656, −10.98120868857743, −10.24934267679741, −9.618899217721730, −8.963294093311753, −8.426741547911679, −8.115475652255809, −7.441828665007727, −6.903629877173823, −6.414039360638518, −5.361680628726776, −4.895738050407536, −4.293151550227792, −3.565466084827268, −3.134354880666772, −2.141412185025383, −1.682855964904129, −0.3576115606910234,
0.3576115606910234, 1.682855964904129, 2.141412185025383, 3.134354880666772, 3.565466084827268, 4.293151550227792, 4.895738050407536, 5.361680628726776, 6.414039360638518, 6.903629877173823, 7.441828665007727, 8.115475652255809, 8.426741547911679, 8.963294093311753, 9.618899217721730, 10.24934267679741, 10.98120868857743, 11.21200878197656, 11.74584765820728, 12.67220187064692, 12.84735990557302, 13.38816891342007, 14.07972480051335, 14.56968845228492, 15.24295049332907
