| L(s) = 1 | + 5-s + 2.37·7-s + 3.37·11-s + 2.37·13-s + 4.74·17-s − 19-s + 0.372·23-s + 25-s − 3.37·29-s + 6.11·31-s + 2.37·35-s + 6·37-s + 11.7·41-s − 6.74·43-s + 3.62·47-s − 1.37·49-s − 7.11·53-s + 3.37·55-s − 5·59-s + 1.25·61-s + 2.37·65-s − 10.7·67-s − 1.37·71-s + 3.25·73-s + 8·77-s − 8.74·79-s + 10·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.896·7-s + 1.01·11-s + 0.657·13-s + 1.15·17-s − 0.229·19-s + 0.0776·23-s + 0.200·25-s − 0.626·29-s + 1.09·31-s + 0.400·35-s + 0.986·37-s + 1.83·41-s − 1.02·43-s + 0.529·47-s − 0.196·49-s − 0.977·53-s + 0.454·55-s − 0.650·59-s + 0.160·61-s + 0.294·65-s − 1.31·67-s − 0.162·71-s + 0.381·73-s + 0.911·77-s − 0.983·79-s + 1.09·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.094815195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.094815195\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 - 2.37T + 7T^{2} \) |
| 11 | \( 1 - 3.37T + 11T^{2} \) |
| 13 | \( 1 - 2.37T + 13T^{2} \) |
| 17 | \( 1 - 4.74T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 0.372T + 23T^{2} \) |
| 29 | \( 1 + 3.37T + 29T^{2} \) |
| 31 | \( 1 - 6.11T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 + 6.74T + 43T^{2} \) |
| 47 | \( 1 - 3.62T + 47T^{2} \) |
| 53 | \( 1 + 7.11T + 53T^{2} \) |
| 59 | \( 1 + 5T + 59T^{2} \) |
| 61 | \( 1 - 1.25T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 1.37T + 71T^{2} \) |
| 73 | \( 1 - 3.25T + 73T^{2} \) |
| 79 | \( 1 + 8.74T + 79T^{2} \) |
| 83 | \( 1 - 10T + 83T^{2} \) |
| 89 | \( 1 - 1.37T + 89T^{2} \) |
| 97 | \( 1 - 6.74T + 97T^{2} \) |
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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
−7.910771313327960846788455889623, −7.50569686030900290527287329541, −6.35276403401918747030593594470, −6.07215948562587193459655444334, −5.14321634669607670616038103209, −4.42373336766855541428134190934, −3.67063152579970817389143199370, −2.72801610236804218385578608138, −1.63753256812706471389335105593, −1.02626896254072784885535362974,
1.02626896254072784885535362974, 1.63753256812706471389335105593, 2.72801610236804218385578608138, 3.67063152579970817389143199370, 4.42373336766855541428134190934, 5.14321634669607670616038103209, 6.07215948562587193459655444334, 6.35276403401918747030593594470, 7.50569686030900290527287329541, 7.910771313327960846788455889623
