| L(s) = 1 | + 2.47·2-s − 0.245·3-s + 4.14·4-s − 2.31·5-s − 0.609·6-s − 0.326·7-s + 5.33·8-s − 2.93·9-s − 5.73·10-s + 0.570·11-s − 1.02·12-s + 0.655·13-s − 0.810·14-s + 0.568·15-s + 4.92·16-s + 1.14·17-s − 7.28·18-s − 2.77·19-s − 9.59·20-s + 0.0803·21-s + 1.41·22-s + 7.23·23-s − 1.31·24-s + 0.347·25-s + 1.62·26-s + 1.46·27-s − 1.35·28-s + ⋯ |
| L(s) = 1 | + 1.75·2-s − 0.141·3-s + 2.07·4-s − 1.03·5-s − 0.248·6-s − 0.123·7-s + 1.88·8-s − 0.979·9-s − 1.81·10-s + 0.172·11-s − 0.294·12-s + 0.181·13-s − 0.216·14-s + 0.146·15-s + 1.23·16-s + 0.277·17-s − 1.71·18-s − 0.635·19-s − 2.14·20-s + 0.0175·21-s + 0.301·22-s + 1.50·23-s − 0.267·24-s + 0.0695·25-s + 0.318·26-s + 0.281·27-s − 0.256·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.173887240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.173887240\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 139 | \( 1 - T \) |
| good | 2 | \( 1 - 2.47T + 2T^{2} \) |
| 3 | \( 1 + 0.245T + 3T^{2} \) |
| 5 | \( 1 + 2.31T + 5T^{2} \) |
| 7 | \( 1 + 0.326T + 7T^{2} \) |
| 11 | \( 1 - 0.570T + 11T^{2} \) |
| 13 | \( 1 - 0.655T + 13T^{2} \) |
| 17 | \( 1 - 1.14T + 17T^{2} \) |
| 19 | \( 1 + 2.77T + 19T^{2} \) |
| 23 | \( 1 - 7.23T + 23T^{2} \) |
| 29 | \( 1 - 7.86T + 29T^{2} \) |
| 31 | \( 1 + 8.91T + 31T^{2} \) |
| 37 | \( 1 + 4.69T + 37T^{2} \) |
| 41 | \( 1 - 9.63T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + 8.96T + 47T^{2} \) |
| 53 | \( 1 - 7.21T + 53T^{2} \) |
| 59 | \( 1 + 5.58T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 3.89T + 67T^{2} \) |
| 71 | \( 1 + 4.74T + 71T^{2} \) |
| 73 | \( 1 + 5.98T + 73T^{2} \) |
| 79 | \( 1 - 6.17T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 + 5.82T + 97T^{2} \) |
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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
−13.07159216399260031145544113082, −12.31868062151420282469624896129, −11.44500324511260075212299729009, −10.83209593578808880254649204403, −8.810106272346937079836710559200, −7.45242522717731671244218278514, −6.31771372530985146828738308465, −5.18798717567746075684120896919, −4.00411638681954913953595376329, −2.91440291277158682988884182729,
2.91440291277158682988884182729, 4.00411638681954913953595376329, 5.18798717567746075684120896919, 6.31771372530985146828738308465, 7.45242522717731671244218278514, 8.810106272346937079836710559200, 10.83209593578808880254649204403, 11.44500324511260075212299729009, 12.31868062151420282469624896129, 13.07159216399260031145544113082
