| L(s) = 1 | + 2.04·2-s + 2.18·4-s + 1.44·5-s − 4.07·7-s + 0.369·8-s + 2.95·10-s − 11-s − 0.740·13-s − 8.32·14-s − 3.60·16-s + 5.60·17-s + 7.10·19-s + 3.15·20-s − 2.04·22-s − 6.31·23-s − 2.90·25-s − 1.51·26-s − 8.87·28-s − 1.15·29-s − 1.07·31-s − 8.11·32-s + 11.4·34-s − 5.89·35-s + 2.50·37-s + 14.5·38-s + 0.534·40-s − 9.71·41-s + ⋯ |
| L(s) = 1 | + 1.44·2-s + 1.09·4-s + 0.647·5-s − 1.53·7-s + 0.130·8-s + 0.935·10-s − 0.301·11-s − 0.205·13-s − 2.22·14-s − 0.901·16-s + 1.35·17-s + 1.62·19-s + 0.705·20-s − 0.435·22-s − 1.31·23-s − 0.581·25-s − 0.297·26-s − 1.67·28-s − 0.214·29-s − 0.193·31-s − 1.43·32-s + 1.96·34-s − 0.995·35-s + 0.411·37-s + 2.35·38-s + 0.0845·40-s − 1.51·41-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 2.04T + 2T^{2} \) |
| 5 | \( 1 - 1.44T + 5T^{2} \) |
| 7 | \( 1 + 4.07T + 7T^{2} \) |
| 13 | \( 1 + 0.740T + 13T^{2} \) |
| 17 | \( 1 - 5.60T + 17T^{2} \) |
| 19 | \( 1 - 7.10T + 19T^{2} \) |
| 23 | \( 1 + 6.31T + 23T^{2} \) |
| 29 | \( 1 + 1.15T + 29T^{2} \) |
| 31 | \( 1 + 1.07T + 31T^{2} \) |
| 37 | \( 1 - 2.50T + 37T^{2} \) |
| 41 | \( 1 + 9.71T + 41T^{2} \) |
| 43 | \( 1 + 8.07T + 43T^{2} \) |
| 47 | \( 1 + 1.61T + 47T^{2} \) |
| 53 | \( 1 + 6.83T + 53T^{2} \) |
| 59 | \( 1 + 3.83T + 59T^{2} \) |
| 67 | \( 1 - 7.40T + 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 + 2.39T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−7.41653331636267657867142153626, −6.74264545457095857407886248703, −5.97614227384999818515583574294, −5.63941887189194388267423813752, −4.98794260388369927959732452019, −3.88449009159091522025095732265, −3.29132662694630861407887986434, −2.81638575417896824476124488188, −1.66292301052291756750933580453, 0,
1.66292301052291756750933580453, 2.81638575417896824476124488188, 3.29132662694630861407887986434, 3.88449009159091522025095732265, 4.98794260388369927959732452019, 5.63941887189194388267423813752, 5.97614227384999818515583574294, 6.74264545457095857407886248703, 7.41653331636267657867142153626
