| L(s) = 1 | + 0.267·5-s − 7-s − 5.19·11-s + 3.46·13-s + 4·17-s + 5.92·19-s − 0.267·23-s − 4.92·25-s − 4.92·29-s + 1.53·31-s − 0.267·35-s − 0.464·37-s + 9.19·41-s − 5.46·43-s + 1.46·47-s + 49-s + 8·53-s − 1.39·55-s − 9.46·59-s + 10.9·61-s + 0.928·65-s + 8.53·67-s − 14.6·71-s + 8.39·73-s + 5.19·77-s + 12.3·79-s + 12.9·83-s + ⋯ |
| L(s) = 1 | + 0.119·5-s − 0.377·7-s − 1.56·11-s + 0.960·13-s + 0.970·17-s + 1.36·19-s − 0.0558·23-s − 0.985·25-s − 0.915·29-s + 0.275·31-s − 0.0452·35-s − 0.0762·37-s + 1.43·41-s − 0.833·43-s + 0.213·47-s + 0.142·49-s + 1.09·53-s − 0.187·55-s − 1.23·59-s + 1.39·61-s + 0.115·65-s + 1.04·67-s − 1.73·71-s + 0.982·73-s + 0.592·77-s + 1.39·79-s + 1.41·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.675636795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.675636795\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 0.267T + 5T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 5.92T + 19T^{2} \) |
| 23 | \( 1 + 0.267T + 23T^{2} \) |
| 29 | \( 1 + 4.92T + 29T^{2} \) |
| 31 | \( 1 - 1.53T + 31T^{2} \) |
| 37 | \( 1 + 0.464T + 37T^{2} \) |
| 41 | \( 1 - 9.19T + 41T^{2} \) |
| 43 | \( 1 + 5.46T + 43T^{2} \) |
| 47 | \( 1 - 1.46T + 47T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 + 9.46T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 8.53T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 - 8.39T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + 1.07T + 97T^{2} \) |
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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
−8.706656130904040624224948701115, −7.70129167661325562264374044191, −7.58626357726489868904360891348, −6.31042754048428183927397628405, −5.61709455052929294764869094618, −5.09179129310354683475812983083, −3.80532246593881169068799261681, −3.15593536102608501060438016228, −2.13300174175086057735239562193, −0.78611957722401249084231990815,
0.78611957722401249084231990815, 2.13300174175086057735239562193, 3.15593536102608501060438016228, 3.80532246593881169068799261681, 5.09179129310354683475812983083, 5.61709455052929294764869094618, 6.31042754048428183927397628405, 7.58626357726489868904360891348, 7.70129167661325562264374044191, 8.706656130904040624224948701115
