| L(s) = 1 | + 0.649·2-s − 1.57·4-s − 3.31·5-s + 1.03·7-s − 2.32·8-s − 2.15·10-s − 11-s − 6.28·13-s + 0.675·14-s + 1.64·16-s − 6.50·17-s − 5.89·19-s + 5.23·20-s − 0.649·22-s − 1.29·23-s + 5.99·25-s − 4.08·26-s − 1.63·28-s − 1.33·29-s − 8.72·31-s + 5.71·32-s − 4.22·34-s − 3.44·35-s − 4.00·37-s − 3.83·38-s + 7.71·40-s + 7.11·41-s + ⋯ |
| L(s) = 1 | + 0.459·2-s − 0.788·4-s − 1.48·5-s + 0.392·7-s − 0.822·8-s − 0.681·10-s − 0.301·11-s − 1.74·13-s + 0.180·14-s + 0.410·16-s − 1.57·17-s − 1.35·19-s + 1.16·20-s − 0.138·22-s − 0.270·23-s + 1.19·25-s − 0.801·26-s − 0.309·28-s − 0.248·29-s − 1.56·31-s + 1.01·32-s − 0.724·34-s − 0.582·35-s − 0.658·37-s − 0.621·38-s + 1.21·40-s + 1.11·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.008044879627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.008044879627\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 0.649T + 2T^{2} \) |
| 5 | \( 1 + 3.31T + 5T^{2} \) |
| 7 | \( 1 - 1.03T + 7T^{2} \) |
| 13 | \( 1 + 6.28T + 13T^{2} \) |
| 17 | \( 1 + 6.50T + 17T^{2} \) |
| 19 | \( 1 + 5.89T + 19T^{2} \) |
| 23 | \( 1 + 1.29T + 23T^{2} \) |
| 29 | \( 1 + 1.33T + 29T^{2} \) |
| 31 | \( 1 + 8.72T + 31T^{2} \) |
| 37 | \( 1 + 4.00T + 37T^{2} \) |
| 41 | \( 1 - 7.11T + 41T^{2} \) |
| 43 | \( 1 + 1.18T + 43T^{2} \) |
| 47 | \( 1 - 4.79T + 47T^{2} \) |
| 53 | \( 1 + 3.64T + 53T^{2} \) |
| 59 | \( 1 + 8.28T + 59T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 - 5.20T + 71T^{2} \) |
| 73 | \( 1 + 16.2T + 73T^{2} \) |
| 79 | \( 1 - 6.50T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 - 8.87T + 89T^{2} \) |
| 97 | \( 1 + 3.27T + 97T^{2} \) |
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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
−7.931489712699123733549370572541, −7.52631308590049660645898300971, −6.76200190896056244128888836563, −5.78074458995588858628073351846, −4.81896275739632935933546141293, −4.48707302058976484242218953043, −3.93557956133906757418563456307, −2.96136335572354204101912500169, −2.02364437059117437823663786924, −0.04117839149477452440013290907,
0.04117839149477452440013290907, 2.02364437059117437823663786924, 2.96136335572354204101912500169, 3.93557956133906757418563456307, 4.48707302058976484242218953043, 4.81896275739632935933546141293, 5.78074458995588858628073351846, 6.76200190896056244128888836563, 7.52631308590049660645898300971, 7.931489712699123733549370572541
