| L(s) = 1 | + 0.618·2-s − 1.61·4-s + 7-s − 2.23·8-s − 3.47·11-s + 6.09·13-s + 0.618·14-s + 1.85·16-s − 6.61·17-s + 0.381·19-s − 2.14·22-s − 4.38·23-s + 3.76·26-s − 1.61·28-s + 2.85·29-s + 3·31-s + 5.61·32-s − 4.09·34-s + 3·37-s + 0.236·38-s − 0.618·41-s + 7.76·43-s + 5.61·44-s − 2.70·46-s − 10.7·47-s + 49-s − 9.85·52-s + ⋯ |
| L(s) = 1 | + 0.437·2-s − 0.809·4-s + 0.377·7-s − 0.790·8-s − 1.04·11-s + 1.68·13-s + 0.165·14-s + 0.463·16-s − 1.60·17-s + 0.0876·19-s − 0.457·22-s − 0.913·23-s + 0.738·26-s − 0.305·28-s + 0.529·29-s + 0.538·31-s + 0.993·32-s − 0.701·34-s + 0.493·37-s + 0.0382·38-s − 0.0965·41-s + 1.18·43-s + 0.846·44-s − 0.399·46-s − 1.56·47-s + 0.142·49-s − 1.36·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.621020306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.621020306\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.618T + 2T^{2} \) |
| 11 | \( 1 + 3.47T + 11T^{2} \) |
| 13 | \( 1 - 6.09T + 13T^{2} \) |
| 17 | \( 1 + 6.61T + 17T^{2} \) |
| 19 | \( 1 - 0.381T + 19T^{2} \) |
| 23 | \( 1 + 4.38T + 23T^{2} \) |
| 29 | \( 1 - 2.85T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 + 0.618T + 41T^{2} \) |
| 43 | \( 1 - 7.76T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 6.61T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 1.14T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 - 8.85T + 71T^{2} \) |
| 73 | \( 1 - 2.52T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−8.268681500946009289345468684616, −7.86929208393732076802756136123, −6.56504056477106466981110945777, −6.09021254005823176396813808602, −5.24524811919679143749375714820, −4.54637875368111877557221348453, −3.93452712835407693228750706818, −3.04047233306564073786069237772, −2.00643455151545433260995511243, −0.65688371009630814879962058150,
0.65688371009630814879962058150, 2.00643455151545433260995511243, 3.04047233306564073786069237772, 3.93452712835407693228750706818, 4.54637875368111877557221348453, 5.24524811919679143749375714820, 6.09021254005823176396813808602, 6.56504056477106466981110945777, 7.86929208393732076802756136123, 8.268681500946009289345468684616
