| L(s) = 1 | + 2-s + 4-s + 2.60·5-s − 0.922·7-s + 8-s + 2.60·10-s + 0.544·11-s − 5.57·13-s − 0.922·14-s + 16-s − 3.06·17-s + 3.41·19-s + 2.60·20-s + 0.544·22-s − 1.81·23-s + 1.80·25-s − 5.57·26-s − 0.922·28-s − 8.11·29-s − 6.23·31-s + 32-s − 3.06·34-s − 2.40·35-s − 0.227·37-s + 3.41·38-s + 2.60·40-s − 5.22·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.16·5-s − 0.348·7-s + 0.353·8-s + 0.825·10-s + 0.164·11-s − 1.54·13-s − 0.246·14-s + 0.250·16-s − 0.743·17-s + 0.784·19-s + 0.583·20-s + 0.116·22-s − 0.379·23-s + 0.361·25-s − 1.09·26-s − 0.174·28-s − 1.50·29-s − 1.11·31-s + 0.176·32-s − 0.526·34-s − 0.407·35-s − 0.0374·37-s + 0.554·38-s + 0.412·40-s − 0.815·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 - 2.60T + 5T^{2} \) |
| 7 | \( 1 + 0.922T + 7T^{2} \) |
| 11 | \( 1 - 0.544T + 11T^{2} \) |
| 13 | \( 1 + 5.57T + 13T^{2} \) |
| 17 | \( 1 + 3.06T + 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 + 1.81T + 23T^{2} \) |
| 29 | \( 1 + 8.11T + 29T^{2} \) |
| 31 | \( 1 + 6.23T + 31T^{2} \) |
| 37 | \( 1 + 0.227T + 37T^{2} \) |
| 41 | \( 1 + 5.22T + 41T^{2} \) |
| 43 | \( 1 + 7.81T + 43T^{2} \) |
| 47 | \( 1 + 0.855T + 47T^{2} \) |
| 53 | \( 1 + 5.74T + 53T^{2} \) |
| 59 | \( 1 - 1.43T + 59T^{2} \) |
| 61 | \( 1 + 8.26T + 61T^{2} \) |
| 67 | \( 1 - 8.90T + 67T^{2} \) |
| 71 | \( 1 + 9.28T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 9.53T + 79T^{2} \) |
| 83 | \( 1 - 2.48T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 2.17T + 97T^{2} \) |
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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
−7.32325796556537156190872579642, −6.63415857309937057181601944650, −6.08737137363527729248370869149, −5.18729744170156279152532917253, −5.00671959667364620977407727822, −3.85540912068992917226152161211, −3.11928158919795793937981071835, −2.17445762736279473930722791843, −1.72675422829293303923482752340, 0,
1.72675422829293303923482752340, 2.17445762736279473930722791843, 3.11928158919795793937981071835, 3.85540912068992917226152161211, 5.00671959667364620977407727822, 5.18729744170156279152532917253, 6.08737137363527729248370869149, 6.63415857309937057181601944650, 7.32325796556537156190872579642
