| L(s) = 1 | − 2.69·2-s + 5.25·4-s − 5-s − 2.74·7-s − 8.76·8-s + 2.69·10-s + 3.38·11-s − 2.46·13-s + 7.38·14-s + 13.1·16-s + 2.64·17-s + 3.38·19-s − 5.25·20-s − 9.12·22-s + 23-s + 25-s + 6.63·26-s − 14.4·28-s − 9.20·29-s − 5.10·31-s − 17.7·32-s − 7.12·34-s + 2.74·35-s + 5.50·37-s − 9.12·38-s + 8.76·40-s + 1.20·41-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 2.62·4-s − 0.447·5-s − 1.03·7-s − 3.09·8-s + 0.851·10-s + 1.02·11-s − 0.683·13-s + 1.97·14-s + 3.27·16-s + 0.641·17-s + 0.777·19-s − 1.17·20-s − 1.94·22-s + 0.208·23-s + 0.200·25-s + 1.30·26-s − 2.72·28-s − 1.70·29-s − 0.917·31-s − 3.14·32-s − 1.22·34-s + 0.463·35-s + 0.904·37-s − 1.47·38-s + 1.38·40-s + 0.188·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 7 | \( 1 + 2.74T + 7T^{2} \) |
| 11 | \( 1 - 3.38T + 11T^{2} \) |
| 13 | \( 1 + 2.46T + 13T^{2} \) |
| 17 | \( 1 - 2.64T + 17T^{2} \) |
| 19 | \( 1 - 3.38T + 19T^{2} \) |
| 29 | \( 1 + 9.20T + 29T^{2} \) |
| 31 | \( 1 + 5.10T + 31T^{2} \) |
| 37 | \( 1 - 5.50T + 37T^{2} \) |
| 41 | \( 1 - 1.20T + 41T^{2} \) |
| 43 | \( 1 - 3.02T + 43T^{2} \) |
| 47 | \( 1 + 8.21T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 + 8.28T + 59T^{2} \) |
| 61 | \( 1 - 0.263T + 61T^{2} \) |
| 67 | \( 1 + 7.66T + 67T^{2} \) |
| 71 | \( 1 - 0.0150T + 71T^{2} \) |
| 73 | \( 1 + 5.53T + 73T^{2} \) |
| 79 | \( 1 + 8.67T + 79T^{2} \) |
| 83 | \( 1 - 3.52T + 83T^{2} \) |
| 89 | \( 1 - 4.66T + 89T^{2} \) |
| 97 | \( 1 + 4.09T + 97T^{2} \) |
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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
−9.407652145642897221537261120838, −9.015998627137688328846387142727, −7.81506847329111350928777685572, −7.31340061162445015985951847296, −6.55764277109648950551944504831, −5.62958374639956538768473228668, −3.74879232965867172154814975152, −2.79708757196334696687636319371, −1.38864983402180821709110777698, 0,
1.38864983402180821709110777698, 2.79708757196334696687636319371, 3.74879232965867172154814975152, 5.62958374639956538768473228668, 6.55764277109648950551944504831, 7.31340061162445015985951847296, 7.81506847329111350928777685572, 9.015998627137688328846387142727, 9.407652145642897221537261120838
