| L(s) = 1 | + 1.61·2-s + 0.618·4-s − 7-s − 2.23·8-s − 4.23·11-s + 3.23·13-s − 1.61·14-s − 4.85·16-s − 6.47·17-s + 4.47·19-s − 6.85·22-s + 1.76·23-s + 5.23·26-s − 0.618·28-s − 5·29-s − 9.70·31-s − 3.38·32-s − 10.4·34-s − 3·37-s + 7.23·38-s − 9.23·41-s − 6.23·43-s − 2.61·44-s + 2.85·46-s − 2·47-s + 49-s + 2.00·52-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.309·4-s − 0.377·7-s − 0.790·8-s − 1.27·11-s + 0.897·13-s − 0.432·14-s − 1.21·16-s − 1.56·17-s + 1.02·19-s − 1.46·22-s + 0.367·23-s + 1.02·26-s − 0.116·28-s − 0.928·29-s − 1.74·31-s − 0.597·32-s − 1.79·34-s − 0.493·37-s + 1.17·38-s − 1.44·41-s − 0.950·43-s − 0.394·44-s + 0.420·46-s − 0.291·47-s + 0.142·49-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 11 | \( 1 + 4.23T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 - 1.76T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 9.70T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 + 9.23T + 41T^{2} \) |
| 43 | \( 1 + 6.23T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 - 1.70T + 59T^{2} \) |
| 61 | \( 1 - 3.70T + 61T^{2} \) |
| 67 | \( 1 + 0.236T + 67T^{2} \) |
| 71 | \( 1 - 4.70T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 5.70T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + 0.763T + 97T^{2} \) |
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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
−9.016674763939395922533721064260, −8.283868080415678386542766332755, −7.15581737363023139572388353792, −6.44572772153427345416446092618, −5.42538953373525037613747107251, −5.04280082945156508818286961776, −3.83932341101615808283304805012, −3.22755375148447618726290811510, −2.09151690798107021125714979894, 0,
2.09151690798107021125714979894, 3.22755375148447618726290811510, 3.83932341101615808283304805012, 5.04280082945156508818286961776, 5.42538953373525037613747107251, 6.44572772153427345416446092618, 7.15581737363023139572388353792, 8.283868080415678386542766332755, 9.016674763939395922533721064260
