| L(s) = 1 | + 1.13·2-s − 1.95·3-s − 0.705·4-s + 1.75·5-s − 2.21·6-s + 1.09·7-s − 3.07·8-s + 0.804·9-s + 1.99·10-s − 11-s + 1.37·12-s + 2.00·13-s + 1.24·14-s − 3.42·15-s − 2.08·16-s + 3.43·17-s + 0.915·18-s − 2.69·19-s − 1.24·20-s − 2.12·21-s − 1.13·22-s + 4.47·23-s + 6.00·24-s − 1.91·25-s + 2.27·26-s + 4.28·27-s − 0.770·28-s + ⋯ |
| L(s) = 1 | + 0.804·2-s − 1.12·3-s − 0.352·4-s + 0.786·5-s − 0.905·6-s + 0.412·7-s − 1.08·8-s + 0.268·9-s + 0.632·10-s − 0.301·11-s + 0.397·12-s + 0.554·13-s + 0.331·14-s − 0.885·15-s − 0.522·16-s + 0.833·17-s + 0.215·18-s − 0.618·19-s − 0.277·20-s − 0.464·21-s − 0.242·22-s + 0.932·23-s + 1.22·24-s − 0.382·25-s + 0.446·26-s + 0.824·27-s − 0.145·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.605154149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.605154149\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 2 | \( 1 - 1.13T + 2T^{2} \) |
| 3 | \( 1 + 1.95T + 3T^{2} \) |
| 5 | \( 1 - 1.75T + 5T^{2} \) |
| 7 | \( 1 - 1.09T + 7T^{2} \) |
| 13 | \( 1 - 2.00T + 13T^{2} \) |
| 17 | \( 1 - 3.43T + 17T^{2} \) |
| 19 | \( 1 + 2.69T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 + 6.09T + 29T^{2} \) |
| 31 | \( 1 - 6.39T + 31T^{2} \) |
| 37 | \( 1 - 9.61T + 37T^{2} \) |
| 41 | \( 1 + 7.70T + 41T^{2} \) |
| 43 | \( 1 - 5.45T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 0.811T + 53T^{2} \) |
| 59 | \( 1 - 15.0T + 59T^{2} \) |
| 61 | \( 1 - 5.64T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 - 5.93T + 71T^{2} \) |
| 73 | \( 1 - 8.69T + 73T^{2} \) |
| 79 | \( 1 - 1.30T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 - 5.83T + 89T^{2} \) |
| 97 | \( 1 - 7.73T + 97T^{2} \) |
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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
−9.641273924434436295557011070763, −8.804994789297759079885890679473, −7.908087174725709689726202598404, −6.62629070191005435984015897693, −5.91550440646656653362016012425, −5.43725774249096755804310951311, −4.73956507869429419882713359804, −3.73058196898316093083941750281, −2.48044204738950198316413617273, −0.870738946478907158830662188961,
0.870738946478907158830662188961, 2.48044204738950198316413617273, 3.73058196898316093083941750281, 4.73956507869429419882713359804, 5.43725774249096755804310951311, 5.91550440646656653362016012425, 6.62629070191005435984015897693, 7.908087174725709689726202598404, 8.804994789297759079885890679473, 9.641273924434436295557011070763
