| L(s) = 1 | − 0.618·2-s + 3-s − 1.61·4-s + 3.23·5-s − 0.618·6-s + 0.236·7-s + 2.23·8-s + 9-s − 2.00·10-s − 0.236·11-s − 1.61·12-s − 5.47·13-s − 0.145·14-s + 3.23·15-s + 1.85·16-s + 3·17-s − 0.618·18-s − 7.23·19-s − 5.23·20-s + 0.236·21-s + 0.145·22-s − 7.70·23-s + 2.23·24-s + 5.47·25-s + 3.38·26-s + 27-s − 0.381·28-s + ⋯ |
| L(s) = 1 | − 0.437·2-s + 0.577·3-s − 0.809·4-s + 1.44·5-s − 0.252·6-s + 0.0892·7-s + 0.790·8-s + 0.333·9-s − 0.632·10-s − 0.0711·11-s − 0.467·12-s − 1.51·13-s − 0.0389·14-s + 0.835·15-s + 0.463·16-s + 0.727·17-s − 0.145·18-s − 1.66·19-s − 1.17·20-s + 0.0515·21-s + 0.0311·22-s − 1.60·23-s + 0.456·24-s + 1.09·25-s + 0.663·26-s + 0.192·27-s − 0.0721·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9366983275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9366983275\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 - 0.236T + 7T^{2} \) |
| 11 | \( 1 + 0.236T + 11T^{2} \) |
| 13 | \( 1 + 5.47T + 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 + 7.70T + 23T^{2} \) |
| 31 | \( 1 - 3.70T + 31T^{2} \) |
| 37 | \( 1 - 5.23T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 4.70T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 + 5.23T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 5.23T + 71T^{2} \) |
| 73 | \( 1 - 6.76T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 2.94T + 83T^{2} \) |
| 89 | \( 1 - 5T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−14.19656138291085195577105289556, −13.30822535943976405292619280405, −12.38690658068640418137094254295, −10.25919111520854853113468939749, −9.848199667372640298808001683822, −8.810338225621425421133394361693, −7.64544354559500736570238313155, −5.91703822599761461769955743526, −4.48583633468729403714404321989, −2.20009654735353763797367045973,
2.20009654735353763797367045973, 4.48583633468729403714404321989, 5.91703822599761461769955743526, 7.64544354559500736570238313155, 8.810338225621425421133394361693, 9.848199667372640298808001683822, 10.25919111520854853113468939749, 12.38690658068640418137094254295, 13.30822535943976405292619280405, 14.19656138291085195577105289556
