| L(s) = 1 | − 0.693·2-s − 0.510·3-s − 1.51·4-s + 0.240·5-s + 0.353·6-s − 7-s + 2.43·8-s − 2.73·9-s − 0.167·10-s − 4.75·11-s + 0.775·12-s − 3.79·13-s + 0.693·14-s − 0.123·15-s + 1.34·16-s + 7.12·17-s + 1.89·18-s + 1.31·19-s − 0.366·20-s + 0.510·21-s + 3.29·22-s − 8.99·23-s − 1.24·24-s − 4.94·25-s + 2.63·26-s + 2.93·27-s + 1.51·28-s + ⋯ |
| L(s) = 1 | − 0.490·2-s − 0.294·3-s − 0.759·4-s + 0.107·5-s + 0.144·6-s − 0.377·7-s + 0.862·8-s − 0.913·9-s − 0.0528·10-s − 1.43·11-s + 0.223·12-s − 1.05·13-s + 0.185·14-s − 0.0317·15-s + 0.337·16-s + 1.72·17-s + 0.447·18-s + 0.301·19-s − 0.0818·20-s + 0.111·21-s + 0.702·22-s − 1.87·23-s − 0.254·24-s − 0.988·25-s + 0.516·26-s + 0.563·27-s + 0.287·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1589 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1589 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4776544938\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4776544938\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 227 | \( 1 - T \) |
| good | 2 | \( 1 + 0.693T + 2T^{2} \) |
| 3 | \( 1 + 0.510T + 3T^{2} \) |
| 5 | \( 1 - 0.240T + 5T^{2} \) |
| 11 | \( 1 + 4.75T + 11T^{2} \) |
| 13 | \( 1 + 3.79T + 13T^{2} \) |
| 17 | \( 1 - 7.12T + 17T^{2} \) |
| 19 | \( 1 - 1.31T + 19T^{2} \) |
| 23 | \( 1 + 8.99T + 23T^{2} \) |
| 29 | \( 1 + 4.12T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 + 4.29T + 37T^{2} \) |
| 41 | \( 1 - 0.427T + 41T^{2} \) |
| 43 | \( 1 - 2.81T + 43T^{2} \) |
| 47 | \( 1 - 4.78T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 - 8.11T + 59T^{2} \) |
| 61 | \( 1 + 6.56T + 61T^{2} \) |
| 67 | \( 1 + 5.90T + 67T^{2} \) |
| 71 | \( 1 + 0.673T + 71T^{2} \) |
| 73 | \( 1 - 3.62T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 9.58T + 83T^{2} \) |
| 89 | \( 1 + 9.96T + 89T^{2} \) |
| 97 | \( 1 - 6.36T + 97T^{2} \) |
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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
−9.672110629077261575169742256426, −8.507009934417578008349359678408, −7.911789463925361437273708940787, −7.37117338110922609367545877934, −5.75375166602508092263900934292, −5.57414888696350545923287527615, −4.51026534594453145345762316021, −3.35184457510693567049696893576, −2.28703786736042906196891649668, −0.50722888874657885410329560947,
0.50722888874657885410329560947, 2.28703786736042906196891649668, 3.35184457510693567049696893576, 4.51026534594453145345762316021, 5.57414888696350545923287527615, 5.75375166602508092263900934292, 7.37117338110922609367545877934, 7.911789463925361437273708940787, 8.507009934417578008349359678408, 9.672110629077261575169742256426
