| L(s) = 1 | + 2.00·2-s − 2.20·3-s + 2.02·4-s − 3.12·5-s − 4.43·6-s + 0.406·7-s + 0.0513·8-s + 1.88·9-s − 6.26·10-s − 4.47·12-s + 1.59·13-s + 0.815·14-s + 6.90·15-s − 3.94·16-s + 0.870·17-s + 3.77·18-s + 7.69·19-s − 6.32·20-s − 0.898·21-s + 5.45·23-s − 0.113·24-s + 4.75·25-s + 3.20·26-s + 2.46·27-s + 0.823·28-s − 9.13·29-s + 13.8·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.27·3-s + 1.01·4-s − 1.39·5-s − 1.80·6-s + 0.153·7-s + 0.0181·8-s + 0.627·9-s − 1.98·10-s − 1.29·12-s + 0.443·13-s + 0.218·14-s + 1.78·15-s − 0.987·16-s + 0.211·17-s + 0.890·18-s + 1.76·19-s − 1.41·20-s − 0.196·21-s + 1.13·23-s − 0.0231·24-s + 0.950·25-s + 0.629·26-s + 0.475·27-s + 0.155·28-s − 1.69·29-s + 2.52·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.619728488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.619728488\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 - 2.00T + 2T^{2} \) |
| 3 | \( 1 + 2.20T + 3T^{2} \) |
| 5 | \( 1 + 3.12T + 5T^{2} \) |
| 7 | \( 1 - 0.406T + 7T^{2} \) |
| 13 | \( 1 - 1.59T + 13T^{2} \) |
| 17 | \( 1 - 0.870T + 17T^{2} \) |
| 19 | \( 1 - 7.69T + 19T^{2} \) |
| 23 | \( 1 - 5.45T + 23T^{2} \) |
| 29 | \( 1 + 9.13T + 29T^{2} \) |
| 31 | \( 1 - 6.93T + 31T^{2} \) |
| 37 | \( 1 - 7.90T + 37T^{2} \) |
| 41 | \( 1 + 2.89T + 41T^{2} \) |
| 43 | \( 1 - 5.29T + 43T^{2} \) |
| 47 | \( 1 - 4.43T + 47T^{2} \) |
| 53 | \( 1 - 4.44T + 53T^{2} \) |
| 59 | \( 1 + 3.39T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 - 5.40T + 67T^{2} \) |
| 71 | \( 1 + 1.81T + 71T^{2} \) |
| 73 | \( 1 - 5.78T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 + 9.70T + 83T^{2} \) |
| 89 | \( 1 + 1.07T + 89T^{2} \) |
| 97 | \( 1 - 9.44T + 97T^{2} \) |
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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
−9.828450402543536146058863913293, −8.709681666313728178704408453295, −7.60536922614031800301945117691, −6.97238313933763178663340928356, −5.96700283590535307932977932474, −5.32856526278037885432356681857, −4.61741834594683669282871880255, −3.79209620473903336489966520394, −2.96894818691429670819715270108, −0.805560398956590894808986292679,
0.805560398956590894808986292679, 2.96894818691429670819715270108, 3.79209620473903336489966520394, 4.61741834594683669282871880255, 5.32856526278037885432356681857, 5.96700283590535307932977932474, 6.97238313933763178663340928356, 7.60536922614031800301945117691, 8.709681666313728178704408453295, 9.828450402543536146058863913293
