L(s) = 1 | + 2-s + 2.04·3-s + 4-s − 5-s + 2.04·6-s − 3.30·7-s + 8-s + 1.18·9-s − 10-s + 4.06·11-s + 2.04·12-s − 6.07·13-s − 3.30·14-s − 2.04·15-s + 16-s − 4.71·17-s + 1.18·18-s + 2.60·19-s − 20-s − 6.76·21-s + 4.06·22-s + 5.83·23-s + 2.04·24-s + 25-s − 6.07·26-s − 3.71·27-s − 3.30·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.18·3-s + 0.5·4-s − 0.447·5-s + 0.835·6-s − 1.25·7-s + 0.353·8-s + 0.394·9-s − 0.316·10-s + 1.22·11-s + 0.590·12-s − 1.68·13-s − 0.884·14-s − 0.528·15-s + 0.250·16-s − 1.14·17-s + 0.278·18-s + 0.598·19-s − 0.223·20-s − 1.47·21-s + 0.865·22-s + 1.21·23-s + 0.417·24-s + 0.200·25-s − 1.19·26-s − 0.714·27-s − 0.625·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 - T \) |
good | 3 | \( 1 - 2.04T + 3T^{2} \) |
| 7 | \( 1 + 3.30T + 7T^{2} \) |
| 11 | \( 1 - 4.06T + 11T^{2} \) |
| 13 | \( 1 + 6.07T + 13T^{2} \) |
| 17 | \( 1 + 4.71T + 17T^{2} \) |
| 19 | \( 1 - 2.60T + 19T^{2} \) |
| 23 | \( 1 - 5.83T + 23T^{2} \) |
| 29 | \( 1 - 3.86T + 29T^{2} \) |
| 31 | \( 1 + 4.92T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 9.93T + 41T^{2} \) |
| 43 | \( 1 - 1.76T + 43T^{2} \) |
| 47 | \( 1 + 1.67T + 47T^{2} \) |
| 53 | \( 1 - 7.93T + 53T^{2} \) |
| 59 | \( 1 + 6.94T + 59T^{2} \) |
| 61 | \( 1 + 8.73T + 61T^{2} \) |
| 67 | \( 1 - 3.39T + 67T^{2} \) |
| 71 | \( 1 + 0.143T + 71T^{2} \) |
| 73 | \( 1 - 1.02T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 2.24T + 97T^{2} \) |
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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
−7.48826609280169081790385773679, −6.96075644374311462343618988318, −6.61087708237105034675640104681, −5.44940150440198960670696411707, −4.65141866416479307612088168931, −3.85180350314377164202802325610, −3.18772820841199522943867862318, −2.72945061635479305136662462663, −1.70984909899190133403846588262, 0,
1.70984909899190133403846588262, 2.72945061635479305136662462663, 3.18772820841199522943867862318, 3.85180350314377164202802325610, 4.65141866416479307612088168931, 5.44940150440198960670696411707, 6.61087708237105034675640104681, 6.96075644374311462343618988318, 7.48826609280169081790385773679