| L(s) = 1 | − 2-s − 3.11·3-s + 4-s + 2.08·5-s + 3.11·6-s − 4.29·7-s − 8-s + 6.70·9-s − 2.08·10-s − 5.59·11-s − 3.11·12-s − 2.42·13-s + 4.29·14-s − 6.49·15-s + 16-s − 1.01·17-s − 6.70·18-s + 5.83·19-s + 2.08·20-s + 13.3·21-s + 5.59·22-s − 23-s + 3.11·24-s − 0.659·25-s + 2.42·26-s − 11.5·27-s − 4.29·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.79·3-s + 0.5·4-s + 0.931·5-s + 1.27·6-s − 1.62·7-s − 0.353·8-s + 2.23·9-s − 0.658·10-s − 1.68·11-s − 0.899·12-s − 0.672·13-s + 1.14·14-s − 1.67·15-s + 0.250·16-s − 0.245·17-s − 1.58·18-s + 1.33·19-s + 0.465·20-s + 2.91·21-s + 1.19·22-s − 0.208·23-s + 0.635·24-s − 0.131·25-s + 0.475·26-s − 2.22·27-s − 0.811·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2915414852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2915414852\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 3.11T + 3T^{2} \) |
| 5 | \( 1 - 2.08T + 5T^{2} \) |
| 7 | \( 1 + 4.29T + 7T^{2} \) |
| 11 | \( 1 + 5.59T + 11T^{2} \) |
| 13 | \( 1 + 2.42T + 13T^{2} \) |
| 17 | \( 1 + 1.01T + 17T^{2} \) |
| 19 | \( 1 - 5.83T + 19T^{2} \) |
| 31 | \( 1 + 3.59T + 31T^{2} \) |
| 37 | \( 1 + 7.58T + 37T^{2} \) |
| 41 | \( 1 + 9.58T + 41T^{2} \) |
| 43 | \( 1 + 0.637T + 43T^{2} \) |
| 47 | \( 1 + 2.11T + 47T^{2} \) |
| 53 | \( 1 - 9.32T + 53T^{2} \) |
| 59 | \( 1 - 3.63T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 5.74T + 67T^{2} \) |
| 71 | \( 1 - 9.47T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 - 5.85T + 89T^{2} \) |
| 97 | \( 1 - 5.12T + 97T^{2} \) |
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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
−9.885991628494566802270458283505, −9.266611889937116479669778964442, −7.75378869437862979082424209618, −6.93696788318870095858798091568, −6.36997126313764636306507608394, −5.45863353673738879717974083889, −5.15390321805852139861913101977, −3.35098233686215685089388782659, −2.10196634266879139529369326590, −0.45034989007500523889645252373,
0.45034989007500523889645252373, 2.10196634266879139529369326590, 3.35098233686215685089388782659, 5.15390321805852139861913101977, 5.45863353673738879717974083889, 6.36997126313764636306507608394, 6.93696788318870095858798091568, 7.75378869437862979082424209618, 9.266611889937116479669778964442, 9.885991628494566802270458283505
