| L(s) = 1 | + 0.618·2-s − 1.61·4-s + 2.61·5-s + 3·7-s − 2.23·8-s + 1.61·10-s + 1.76·13-s + 1.85·14-s + 1.85·16-s − 1.61·17-s + 5.85·19-s − 4.23·20-s − 3.47·23-s + 1.85·25-s + 1.09·26-s − 4.85·28-s − 4.47·29-s + 2.85·31-s + 5.61·32-s − 1.00·34-s + 7.85·35-s + 0.236·37-s + 3.61·38-s − 5.85·40-s + 11.9·41-s + 6.23·43-s − 2.14·46-s + ⋯ |
| L(s) = 1 | + 0.437·2-s − 0.809·4-s + 1.17·5-s + 1.13·7-s − 0.790·8-s + 0.511·10-s + 0.489·13-s + 0.495·14-s + 0.463·16-s − 0.392·17-s + 1.34·19-s − 0.947·20-s − 0.723·23-s + 0.370·25-s + 0.213·26-s − 0.917·28-s − 0.830·29-s + 0.512·31-s + 0.993·32-s − 0.171·34-s + 1.32·35-s + 0.0388·37-s + 0.586·38-s − 0.925·40-s + 1.86·41-s + 0.950·43-s − 0.316·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.303607830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.303607830\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 5 | \( 1 - 2.61T + 5T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 17 | \( 1 + 1.61T + 17T^{2} \) |
| 19 | \( 1 - 5.85T + 19T^{2} \) |
| 23 | \( 1 + 3.47T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 2.85T + 31T^{2} \) |
| 37 | \( 1 - 0.236T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 - 6.23T + 43T^{2} \) |
| 47 | \( 1 + 1.61T + 47T^{2} \) |
| 53 | \( 1 - 9.61T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 7.85T + 61T^{2} \) |
| 67 | \( 1 + 9.56T + 67T^{2} \) |
| 71 | \( 1 - 5.56T + 71T^{2} \) |
| 73 | \( 1 + 3.23T + 73T^{2} \) |
| 79 | \( 1 - 9.47T + 79T^{2} \) |
| 83 | \( 1 + 0.708T + 83T^{2} \) |
| 89 | \( 1 + 0.527T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−9.669675606154122978217542182674, −9.205434234410258669521308365187, −8.292843353602385998952492378525, −7.47633320726853152340356988229, −6.07244157986135582478456731306, −5.58126377552550788542594857978, −4.74953947796879796162142854043, −3.83717781916700393658507381675, −2.46886049989690408429082169423, −1.22954921924342492536043808731,
1.22954921924342492536043808731, 2.46886049989690408429082169423, 3.83717781916700393658507381675, 4.74953947796879796162142854043, 5.58126377552550788542594857978, 6.07244157986135582478456731306, 7.47633320726853152340356988229, 8.292843353602385998952492378525, 9.205434234410258669521308365187, 9.669675606154122978217542182674
