L(s) = 1 | + 1.02·3-s − 0.697·5-s + 2.22·7-s − 1.95·9-s − 2.12·11-s + 5.48·13-s − 0.713·15-s − 17-s + 3.75·19-s + 2.28·21-s − 8.58·23-s − 4.51·25-s − 5.07·27-s + 6.09·29-s + 7.28·31-s − 2.17·33-s − 1.55·35-s + 8.14·37-s + 5.61·39-s − 9.31·41-s + 1.78·43-s + 1.35·45-s + 8.37·47-s − 2.04·49-s − 1.02·51-s + 3.87·53-s + 1.48·55-s + ⋯ |
L(s) = 1 | + 0.591·3-s − 0.311·5-s + 0.841·7-s − 0.650·9-s − 0.640·11-s + 1.52·13-s − 0.184·15-s − 0.242·17-s + 0.861·19-s + 0.497·21-s − 1.78·23-s − 0.902·25-s − 0.975·27-s + 1.13·29-s + 1.30·31-s − 0.378·33-s − 0.262·35-s + 1.33·37-s + 0.899·39-s − 1.45·41-s + 0.271·43-s + 0.202·45-s + 1.22·47-s − 0.291·49-s − 0.143·51-s + 0.532·53-s + 0.199·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.494492800\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.494492800\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 - 1.02T + 3T^{2} \) |
| 5 | \( 1 + 0.697T + 5T^{2} \) |
| 7 | \( 1 - 2.22T + 7T^{2} \) |
| 11 | \( 1 + 2.12T + 11T^{2} \) |
| 13 | \( 1 - 5.48T + 13T^{2} \) |
| 19 | \( 1 - 3.75T + 19T^{2} \) |
| 23 | \( 1 + 8.58T + 23T^{2} \) |
| 29 | \( 1 - 6.09T + 29T^{2} \) |
| 31 | \( 1 - 7.28T + 31T^{2} \) |
| 37 | \( 1 - 8.14T + 37T^{2} \) |
| 41 | \( 1 + 9.31T + 41T^{2} \) |
| 43 | \( 1 - 1.78T + 43T^{2} \) |
| 47 | \( 1 - 8.37T + 47T^{2} \) |
| 53 | \( 1 - 3.87T + 53T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 0.203T + 67T^{2} \) |
| 71 | \( 1 + 9.31T + 71T^{2} \) |
| 73 | \( 1 - 2.79T + 73T^{2} \) |
| 79 | \( 1 + 0.630T + 79T^{2} \) |
| 83 | \( 1 - 0.0496T + 83T^{2} \) |
| 89 | \( 1 + 2.56T + 89T^{2} \) |
| 97 | \( 1 - 5.53T + 97T^{2} \) |
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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
−7.996013196190313229383114101887, −7.45529725827763207320517753130, −6.24770897208290226558755240680, −5.88794120156130148404410543444, −4.98695964790201643493955556279, −4.16726772066906715874721220778, −3.52781646620863737381409985772, −2.66642751364343539931968214111, −1.88683398651552488968014796278, −0.76245305755614858921776938722,
0.76245305755614858921776938722, 1.88683398651552488968014796278, 2.66642751364343539931968214111, 3.52781646620863737381409985772, 4.16726772066906715874721220778, 4.98695964790201643493955556279, 5.88794120156130148404410543444, 6.24770897208290226558755240680, 7.45529725827763207320517753130, 7.996013196190313229383114101887