| L(s) = 1 | − 0.618·2-s − 1.61·4-s − 5-s − 7-s + 2.23·8-s + 0.618·10-s + 0.236·11-s + 3.61·13-s + 0.618·14-s + 1.85·16-s + 0.618·17-s + 4.09·19-s + 1.61·20-s − 0.145·22-s − 7.61·23-s + 25-s − 2.23·26-s + 1.61·28-s − 10.5·29-s − 6.70·31-s − 5.61·32-s − 0.381·34-s + 35-s − 6.70·37-s − 2.52·38-s − 2.23·40-s + 3.09·41-s + ⋯ |
| L(s) = 1 | − 0.437·2-s − 0.809·4-s − 0.447·5-s − 0.377·7-s + 0.790·8-s + 0.195·10-s + 0.0711·11-s + 1.00·13-s + 0.165·14-s + 0.463·16-s + 0.149·17-s + 0.938·19-s + 0.361·20-s − 0.0311·22-s − 1.58·23-s + 0.200·25-s − 0.438·26-s + 0.305·28-s − 1.96·29-s − 1.20·31-s − 0.993·32-s − 0.0655·34-s + 0.169·35-s − 1.10·37-s − 0.410·38-s − 0.353·40-s + 0.482·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 11 | \( 1 - 0.236T + 11T^{2} \) |
| 13 | \( 1 - 3.61T + 13T^{2} \) |
| 17 | \( 1 - 0.618T + 17T^{2} \) |
| 19 | \( 1 - 4.09T + 19T^{2} \) |
| 23 | \( 1 + 7.61T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 + 6.70T + 37T^{2} \) |
| 41 | \( 1 - 3.09T + 41T^{2} \) |
| 43 | \( 1 - 7.94T + 43T^{2} \) |
| 47 | \( 1 + 11T + 47T^{2} \) |
| 53 | \( 1 - 6.61T + 53T^{2} \) |
| 59 | \( 1 + 14.7T + 59T^{2} \) |
| 61 | \( 1 + 7.14T + 61T^{2} \) |
| 67 | \( 1 + 4.32T + 67T^{2} \) |
| 71 | \( 1 - 2.85T + 71T^{2} \) |
| 73 | \( 1 + 3.94T + 73T^{2} \) |
| 79 | \( 1 - 3.38T + 79T^{2} \) |
| 83 | \( 1 - 2.70T + 83T^{2} \) |
| 89 | \( 1 - 7.47T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−9.406703153240767885191554916019, −8.991227256322384876700815324382, −7.920222104281531654170329335234, −7.44388713694371732256281233477, −6.10549484652247078006660884337, −5.29129357206834833475015060053, −4.03079214007870853271298107808, −3.47150343482780152789955580828, −1.58618687098928773813721097511, 0,
1.58618687098928773813721097511, 3.47150343482780152789955580828, 4.03079214007870853271298107808, 5.29129357206834833475015060053, 6.10549484652247078006660884337, 7.44388713694371732256281233477, 7.920222104281531654170329335234, 8.991227256322384876700815324382, 9.406703153240767885191554916019
