| L(s) = 1 | − 0.731·2-s − 1.46·4-s − 2.76·5-s − 1.34·7-s + 2.53·8-s + 2.02·10-s + 11-s − 4.81·13-s + 0.985·14-s + 1.07·16-s − 6.15·17-s − 0.496·19-s + 4.04·20-s − 0.731·22-s − 7.44·23-s + 2.64·25-s + 3.52·26-s + 1.97·28-s + 0.647·29-s + 4.67·31-s − 5.85·32-s + 4.50·34-s + 3.72·35-s + 11.3·37-s + 0.363·38-s − 7.00·40-s − 9.09·41-s + ⋯ |
| L(s) = 1 | − 0.517·2-s − 0.732·4-s − 1.23·5-s − 0.508·7-s + 0.896·8-s + 0.639·10-s + 0.301·11-s − 1.33·13-s + 0.263·14-s + 0.268·16-s − 1.49·17-s − 0.113·19-s + 0.905·20-s − 0.156·22-s − 1.55·23-s + 0.528·25-s + 0.690·26-s + 0.372·28-s + 0.120·29-s + 0.840·31-s − 1.03·32-s + 0.772·34-s + 0.629·35-s + 1.86·37-s + 0.0589·38-s − 1.10·40-s − 1.42·41-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
| \(L(1)\) |
\(\approx\) |
\(0.001694683644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.001694683644\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.731T + 2T^{2} \) |
| 5 | \( 1 + 2.76T + 5T^{2} \) |
| 7 | \( 1 + 1.34T + 7T^{2} \) |
| 13 | \( 1 + 4.81T + 13T^{2} \) |
| 17 | \( 1 + 6.15T + 17T^{2} \) |
| 19 | \( 1 + 0.496T + 19T^{2} \) |
| 23 | \( 1 + 7.44T + 23T^{2} \) |
| 29 | \( 1 - 0.647T + 29T^{2} \) |
| 31 | \( 1 - 4.67T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 + 9.09T + 41T^{2} \) |
| 43 | \( 1 + 9.64T + 43T^{2} \) |
| 47 | \( 1 + 8.36T + 47T^{2} \) |
| 53 | \( 1 + 9.40T + 53T^{2} \) |
| 59 | \( 1 + 0.756T + 59T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 4.46T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 2.10T + 97T^{2} \) |
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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
−8.152808456839626216351773454686, −7.58714612121916832749406158098, −6.80403599214215095165017840144, −6.13168775459669027643133103847, −4.78390094786723218042903290312, −4.53949873533596493023086887388, −3.77745014623490563079066742714, −2.85527895210899338449715106620, −1.67959133434723088305258204675, −0.02300495767575888043066198606,
0.02300495767575888043066198606, 1.67959133434723088305258204675, 2.85527895210899338449715106620, 3.77745014623490563079066742714, 4.53949873533596493023086887388, 4.78390094786723218042903290312, 6.13168775459669027643133103847, 6.80403599214215095165017840144, 7.58714612121916832749406158098, 8.152808456839626216351773454686
