| L(s) = 1 | + 0.934·2-s − 3-s − 1.12·4-s + 0.782·5-s − 0.934·6-s − 7-s − 2.92·8-s + 9-s + 0.731·10-s − 3.89·11-s + 1.12·12-s − 4.44·13-s − 0.934·14-s − 0.782·15-s − 0.475·16-s + 3.09·17-s + 0.934·18-s + 2.66·19-s − 0.882·20-s + 21-s − 3.64·22-s − 6.01·23-s + 2.92·24-s − 4.38·25-s − 4.15·26-s − 27-s + 1.12·28-s + ⋯ |
| L(s) = 1 | + 0.660·2-s − 0.577·3-s − 0.563·4-s + 0.349·5-s − 0.381·6-s − 0.377·7-s − 1.03·8-s + 0.333·9-s + 0.231·10-s − 1.17·11-s + 0.325·12-s − 1.23·13-s − 0.249·14-s − 0.202·15-s − 0.118·16-s + 0.750·17-s + 0.220·18-s + 0.611·19-s − 0.197·20-s + 0.218·21-s − 0.776·22-s − 1.25·23-s + 0.596·24-s − 0.877·25-s − 0.815·26-s − 0.192·27-s + 0.212·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.008856560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.008856560\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 - T \) |
| good | 2 | \( 1 - 0.934T + 2T^{2} \) |
| 5 | \( 1 - 0.782T + 5T^{2} \) |
| 11 | \( 1 + 3.89T + 11T^{2} \) |
| 13 | \( 1 + 4.44T + 13T^{2} \) |
| 17 | \( 1 - 3.09T + 17T^{2} \) |
| 19 | \( 1 - 2.66T + 19T^{2} \) |
| 23 | \( 1 + 6.01T + 23T^{2} \) |
| 29 | \( 1 - 0.393T + 29T^{2} \) |
| 31 | \( 1 - 6.54T + 31T^{2} \) |
| 37 | \( 1 + 0.294T + 37T^{2} \) |
| 41 | \( 1 - 6.19T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 3.92T + 47T^{2} \) |
| 53 | \( 1 + 4.58T + 53T^{2} \) |
| 59 | \( 1 - 2.49T + 59T^{2} \) |
| 61 | \( 1 - 6.93T + 61T^{2} \) |
| 67 | \( 1 - 9.15T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 6.42T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + 7.80T + 83T^{2} \) |
| 89 | \( 1 + 4.54T + 89T^{2} \) |
| 97 | \( 1 - 4.54T + 97T^{2} \) |
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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
−8.190195777233688048762924051026, −7.83192652399723972393946857676, −6.75561218668008177151216447480, −6.00152664244598886313315098422, −5.29973192181312252937258754121, −4.94394208476279352913611820430, −3.96716510987707523062491643630, −3.07536360152997967397973431355, −2.17811565181590653360260514763, −0.51240592589547072013328576652,
0.51240592589547072013328576652, 2.17811565181590653360260514763, 3.07536360152997967397973431355, 3.96716510987707523062491643630, 4.94394208476279352913611820430, 5.29973192181312252937258754121, 6.00152664244598886313315098422, 6.75561218668008177151216447480, 7.83192652399723972393946857676, 8.190195777233688048762924051026
