| L(s) = 1 | + 2.37·2-s − 3-s + 3.63·4-s − 3.39·5-s − 2.37·6-s − 2.72·7-s + 3.87·8-s + 9-s − 8.04·10-s + 0.191·11-s − 3.63·12-s + 3.63·13-s − 6.46·14-s + 3.39·15-s + 1.92·16-s − 17-s + 2.37·18-s + 6.06·19-s − 12.3·20-s + 2.72·21-s + 0.455·22-s + 7.64·23-s − 3.87·24-s + 6.49·25-s + 8.61·26-s − 27-s − 9.89·28-s + ⋯ |
| L(s) = 1 | + 1.67·2-s − 0.577·3-s + 1.81·4-s − 1.51·5-s − 0.968·6-s − 1.02·7-s + 1.36·8-s + 0.333·9-s − 2.54·10-s + 0.0578·11-s − 1.04·12-s + 1.00·13-s − 1.72·14-s + 0.875·15-s + 0.481·16-s − 0.242·17-s + 0.559·18-s + 1.39·19-s − 2.75·20-s + 0.594·21-s + 0.0970·22-s + 1.59·23-s − 0.790·24-s + 1.29·25-s + 1.69·26-s − 0.192·27-s − 1.87·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 - 2.37T + 2T^{2} \) |
| 5 | \( 1 + 3.39T + 5T^{2} \) |
| 7 | \( 1 + 2.72T + 7T^{2} \) |
| 11 | \( 1 - 0.191T + 11T^{2} \) |
| 13 | \( 1 - 3.63T + 13T^{2} \) |
| 19 | \( 1 - 6.06T + 19T^{2} \) |
| 23 | \( 1 - 7.64T + 23T^{2} \) |
| 29 | \( 1 + 5.48T + 29T^{2} \) |
| 31 | \( 1 + 6.85T + 31T^{2} \) |
| 37 | \( 1 - 2.95T + 37T^{2} \) |
| 41 | \( 1 - 5.70T + 41T^{2} \) |
| 43 | \( 1 + 1.32T + 43T^{2} \) |
| 47 | \( 1 - 2.64T + 47T^{2} \) |
| 53 | \( 1 + 5.38T + 53T^{2} \) |
| 59 | \( 1 + 7.04T + 59T^{2} \) |
| 61 | \( 1 + 1.28T + 61T^{2} \) |
| 67 | \( 1 + 8.81T + 67T^{2} \) |
| 71 | \( 1 + 6.29T + 71T^{2} \) |
| 73 | \( 1 + 7.91T + 73T^{2} \) |
| 79 | \( 1 + 5.71T + 79T^{2} \) |
| 83 | \( 1 + 0.273T + 83T^{2} \) |
| 89 | \( 1 - 8.77T + 89T^{2} \) |
| 97 | \( 1 - 3.56T + 97T^{2} \) |
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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
−7.36786792240977481887655299716, −6.59784940071398818846369578017, −5.97903873954848889089006105188, −5.32361559639611559793409414533, −4.58421411917827919464853310692, −3.88235605918528126482582817051, −3.38897432784492683025780225483, −2.87909900689215997853149673030, −1.29520717741806381173847091008, 0,
1.29520717741806381173847091008, 2.87909900689215997853149673030, 3.38897432784492683025780225483, 3.88235605918528126482582817051, 4.58421411917827919464853310692, 5.32361559639611559793409414533, 5.97903873954848889089006105188, 6.59784940071398818846369578017, 7.36786792240977481887655299716
