| L(s) = 1 | − 2-s − 3.18·3-s + 4-s − 0.979·5-s + 3.18·6-s + 4.69·7-s − 8-s + 7.14·9-s + 0.979·10-s + 3.33·11-s − 3.18·12-s + 5.77·13-s − 4.69·14-s + 3.12·15-s + 16-s − 2.62·17-s − 7.14·18-s + 7.07·19-s − 0.979·20-s − 14.9·21-s − 3.33·22-s − 23-s + 3.18·24-s − 4.03·25-s − 5.77·26-s − 13.2·27-s + 4.69·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.83·3-s + 0.5·4-s − 0.438·5-s + 1.30·6-s + 1.77·7-s − 0.353·8-s + 2.38·9-s + 0.309·10-s + 1.00·11-s − 0.919·12-s + 1.60·13-s − 1.25·14-s + 0.806·15-s + 0.250·16-s − 0.636·17-s − 1.68·18-s + 1.62·19-s − 0.219·20-s − 3.26·21-s − 0.710·22-s − 0.208·23-s + 0.650·24-s − 0.807·25-s − 1.13·26-s − 2.54·27-s + 0.887·28-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
| \(L(1)\) |
\(\approx\) |
\(0.8931853244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8931853244\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.18T + 3T^{2} \) |
| 5 | \( 1 + 0.979T + 5T^{2} \) |
| 7 | \( 1 - 4.69T + 7T^{2} \) |
| 11 | \( 1 - 3.33T + 11T^{2} \) |
| 13 | \( 1 - 5.77T + 13T^{2} \) |
| 17 | \( 1 + 2.62T + 17T^{2} \) |
| 19 | \( 1 - 7.07T + 19T^{2} \) |
| 31 | \( 1 - 5.33T + 31T^{2} \) |
| 37 | \( 1 - 3.47T + 37T^{2} \) |
| 41 | \( 1 - 1.47T + 41T^{2} \) |
| 43 | \( 1 + 9.70T + 43T^{2} \) |
| 47 | \( 1 + 7.25T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 - 1.40T + 59T^{2} \) |
| 61 | \( 1 + 0.994T + 61T^{2} \) |
| 67 | \( 1 + 0.381T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 5.47T + 83T^{2} \) |
| 89 | \( 1 + 5.89T + 89T^{2} \) |
| 97 | \( 1 + 3.33T + 97T^{2} \) |
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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
−9.829179932227422228935304760291, −8.714281215500180668938902601495, −7.976833920231902867607961740057, −7.13546547132862041482595211803, −6.31083025818117194806927397453, −5.56706465034709142217447002794, −4.67195965168604991728636326656, −3.85340910040553450493425644834, −1.58462708458992996973490060953, −0.969965419987869845313287018566,
0.969965419987869845313287018566, 1.58462708458992996973490060953, 3.85340910040553450493425644834, 4.67195965168604991728636326656, 5.56706465034709142217447002794, 6.31083025818117194806927397453, 7.13546547132862041482595211803, 7.976833920231902867607961740057, 8.714281215500180668938902601495, 9.829179932227422228935304760291
