L(s) = 1 | + 3.46·5-s + 2.73·7-s − 1.26·11-s + 5.46·13-s + 17-s + 1.46·19-s − 1.26·23-s + 6.99·25-s − 3.46·29-s − 4.19·31-s + 9.46·35-s + 4.53·37-s + 6·41-s + 8.39·43-s − 6.92·47-s + 0.464·49-s − 12.9·53-s − 4.39·55-s + 2.53·59-s − 0.535·61-s + 18.9·65-s − 14.9·67-s − 8.19·71-s + 2·73-s − 3.46·77-s + 12.1·79-s − 2.53·83-s + ⋯ |
L(s) = 1 | + 1.54·5-s + 1.03·7-s − 0.382·11-s + 1.51·13-s + 0.242·17-s + 0.335·19-s − 0.264·23-s + 1.39·25-s − 0.643·29-s − 0.753·31-s + 1.59·35-s + 0.745·37-s + 0.937·41-s + 1.27·43-s − 1.01·47-s + 0.0663·49-s − 1.77·53-s − 0.592·55-s + 0.330·59-s − 0.0686·61-s + 2.34·65-s − 1.82·67-s − 0.972·71-s + 0.234·73-s − 0.394·77-s + 1.37·79-s − 0.278·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.965032416\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.965032416\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 13 | \( 1 - 5.46T + 13T^{2} \) |
| 19 | \( 1 - 1.46T + 19T^{2} \) |
| 23 | \( 1 + 1.26T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 4.19T + 31T^{2} \) |
| 37 | \( 1 - 4.53T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 8.39T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 2.53T + 59T^{2} \) |
| 61 | \( 1 + 0.535T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 8.19T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 2.53T + 83T^{2} \) |
| 89 | \( 1 + 2.53T + 89T^{2} \) |
| 97 | \( 1 + 4.92T + 97T^{2} \) |
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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
−9.103531827427026962448751478494, −8.169835331724002345155829463575, −7.54836850473106401720466531845, −6.35015911811647586892199197286, −5.83687246827700899302500374557, −5.19423728804005685528031419392, −4.22154470235756127684848620732, −3.04232869927764680838215700614, −1.93691368932180993169942848671, −1.27114499707278190804568519501,
1.27114499707278190804568519501, 1.93691368932180993169942848671, 3.04232869927764680838215700614, 4.22154470235756127684848620732, 5.19423728804005685528031419392, 5.83687246827700899302500374557, 6.35015911811647586892199197286, 7.54836850473106401720466531845, 8.169835331724002345155829463575, 9.103531827427026962448751478494