| L(s) = 1 | + 3-s + 1.63·5-s − 0.577·7-s + 9-s + 5.51·11-s + 0.530·13-s + 1.63·15-s + 6.30·17-s + 0.869·19-s − 0.577·21-s + 23-s − 2.34·25-s + 27-s − 29-s − 4.68·31-s + 5.51·33-s − 0.941·35-s − 2.51·37-s + 0.530·39-s + 7.33·41-s + 7.82·43-s + 1.63·45-s + 4.94·47-s − 6.66·49-s + 6.30·51-s + 2.53·53-s + 8.99·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.728·5-s − 0.218·7-s + 0.333·9-s + 1.66·11-s + 0.147·13-s + 0.420·15-s + 1.52·17-s + 0.199·19-s − 0.126·21-s + 0.208·23-s − 0.468·25-s + 0.192·27-s − 0.185·29-s − 0.840·31-s + 0.960·33-s − 0.159·35-s − 0.414·37-s + 0.0848·39-s + 1.14·41-s + 1.19·43-s + 0.242·45-s + 0.721·47-s − 0.952·49-s + 0.883·51-s + 0.348·53-s + 1.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.719388802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.719388802\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 1.63T + 5T^{2} \) |
| 7 | \( 1 + 0.577T + 7T^{2} \) |
| 11 | \( 1 - 5.51T + 11T^{2} \) |
| 13 | \( 1 - 0.530T + 13T^{2} \) |
| 17 | \( 1 - 6.30T + 17T^{2} \) |
| 19 | \( 1 - 0.869T + 19T^{2} \) |
| 31 | \( 1 + 4.68T + 31T^{2} \) |
| 37 | \( 1 + 2.51T + 37T^{2} \) |
| 41 | \( 1 - 7.33T + 41T^{2} \) |
| 43 | \( 1 - 7.82T + 43T^{2} \) |
| 47 | \( 1 - 4.94T + 47T^{2} \) |
| 53 | \( 1 - 2.53T + 53T^{2} \) |
| 59 | \( 1 - 14.0T + 59T^{2} \) |
| 61 | \( 1 + 5.51T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 9.56T + 71T^{2} \) |
| 73 | \( 1 + 3.25T + 73T^{2} \) |
| 79 | \( 1 - 0.347T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 - 2.03T + 97T^{2} \) |
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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
−7.64056190935897453511080817401, −7.33281687794757748950948333520, −6.30696350075628989789053458923, −5.92686981145048990162068997783, −5.10924164426307824879037252386, −4.00806065561990963619132795847, −3.60757675599384505485592016635, −2.67917047695031923373062108827, −1.70764937931677192887086454240, −1.02082646472062891868015763374,
1.02082646472062891868015763374, 1.70764937931677192887086454240, 2.67917047695031923373062108827, 3.60757675599384505485592016635, 4.00806065561990963619132795847, 5.10924164426307824879037252386, 5.92686981145048990162068997783, 6.30696350075628989789053458923, 7.33281687794757748950948333520, 7.64056190935897453511080817401
