| L(s) = 1 | + 2.57·2-s + 4.64·4-s − 2.12·5-s − 3.45·7-s + 6.80·8-s − 5.48·10-s + 11-s − 2.79·13-s − 8.90·14-s + 8.25·16-s + 6.16·17-s − 2.59·19-s − 9.86·20-s + 2.57·22-s − 0.281·23-s − 0.476·25-s − 7.20·26-s − 16.0·28-s + 0.776·29-s + 3.56·31-s + 7.65·32-s + 15.8·34-s + 7.34·35-s − 11.5·37-s − 6.68·38-s − 14.4·40-s − 10.0·41-s + ⋯ |
| L(s) = 1 | + 1.82·2-s + 2.32·4-s − 0.951·5-s − 1.30·7-s + 2.40·8-s − 1.73·10-s + 0.301·11-s − 0.775·13-s − 2.37·14-s + 2.06·16-s + 1.49·17-s − 0.594·19-s − 2.20·20-s + 0.549·22-s − 0.0587·23-s − 0.0952·25-s − 1.41·26-s − 3.02·28-s + 0.144·29-s + 0.640·31-s + 1.35·32-s + 2.72·34-s + 1.24·35-s − 1.89·37-s − 1.08·38-s − 2.28·40-s − 1.57·41-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 | \( 1 - 2.57T + 2T^{2} \) |
| 5 | \( 1 + 2.12T + 5T^{2} \) |
| 7 | \( 1 + 3.45T + 7T^{2} \) |
| 13 | \( 1 + 2.79T + 13T^{2} \) |
| 17 | \( 1 - 6.16T + 17T^{2} \) |
| 19 | \( 1 + 2.59T + 19T^{2} \) |
| 23 | \( 1 + 0.281T + 23T^{2} \) |
| 29 | \( 1 - 0.776T + 29T^{2} \) |
| 31 | \( 1 - 3.56T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 4.63T + 43T^{2} \) |
| 47 | \( 1 - 0.572T + 47T^{2} \) |
| 53 | \( 1 - 1.09T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 + 0.0926T + 71T^{2} \) |
| 73 | \( 1 + 3.82T + 73T^{2} \) |
| 79 | \( 1 + 8.59T + 79T^{2} \) |
| 83 | \( 1 - 2.92T + 83T^{2} \) |
| 89 | \( 1 - 5.34T + 89T^{2} \) |
| 97 | \( 1 - 4.05T + 97T^{2} \) |
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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
−7.36976926172322384935781321583, −6.81026024121626582491201576305, −6.22400009591113007977209108438, −5.45352025384730562110393846679, −4.75034597464606239476950544840, −3.96779041257488235362684488601, −3.32221992414224201770610013112, −2.97226179990672158094210373848, −1.72126055484203674876469300344, 0,
1.72126055484203674876469300344, 2.97226179990672158094210373848, 3.32221992414224201770610013112, 3.96779041257488235362684488601, 4.75034597464606239476950544840, 5.45352025384730562110393846679, 6.22400009591113007977209108438, 6.81026024121626582491201576305, 7.36976926172322384935781321583
