| L(s) = 1 | + 0.356·3-s − 5-s − 0.400·7-s − 2.87·9-s − 0.436·11-s − 0.615·13-s − 0.356·15-s − 0.540·17-s + 1.98·19-s − 0.142·21-s + 0.0321·23-s + 25-s − 2.09·27-s + 7.32·29-s − 1.85·31-s − 0.155·33-s + 0.400·35-s + 1.34·37-s − 0.219·39-s + 7.68·41-s + 4.12·43-s + 2.87·45-s + 11.3·47-s − 6.83·49-s − 0.192·51-s − 3.77·53-s + 0.436·55-s + ⋯ |
| L(s) = 1 | + 0.205·3-s − 0.447·5-s − 0.151·7-s − 0.957·9-s − 0.131·11-s − 0.170·13-s − 0.0919·15-s − 0.131·17-s + 0.454·19-s − 0.0311·21-s + 0.00669·23-s + 0.200·25-s − 0.402·27-s + 1.36·29-s − 0.332·31-s − 0.0270·33-s + 0.0676·35-s + 0.220·37-s − 0.0350·39-s + 1.20·41-s + 0.629·43-s + 0.428·45-s + 1.64·47-s − 0.977·49-s − 0.0269·51-s − 0.518·53-s + 0.0588·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 0.356T + 3T^{2} \) |
| 7 | \( 1 + 0.400T + 7T^{2} \) |
| 11 | \( 1 + 0.436T + 11T^{2} \) |
| 13 | \( 1 + 0.615T + 13T^{2} \) |
| 17 | \( 1 + 0.540T + 17T^{2} \) |
| 19 | \( 1 - 1.98T + 19T^{2} \) |
| 23 | \( 1 - 0.0321T + 23T^{2} \) |
| 29 | \( 1 - 7.32T + 29T^{2} \) |
| 31 | \( 1 + 1.85T + 31T^{2} \) |
| 37 | \( 1 - 1.34T + 37T^{2} \) |
| 41 | \( 1 - 7.68T + 41T^{2} \) |
| 43 | \( 1 - 4.12T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 3.77T + 53T^{2} \) |
| 59 | \( 1 - 3.82T + 59T^{2} \) |
| 61 | \( 1 + 1.59T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 8.53T + 71T^{2} \) |
| 73 | \( 1 - 1.94T + 73T^{2} \) |
| 79 | \( 1 + 2.05T + 79T^{2} \) |
| 83 | \( 1 - 3.42T + 83T^{2} \) |
| 89 | \( 1 + 9.88T + 89T^{2} \) |
| 97 | \( 1 + 4.60T + 97T^{2} \) |
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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
−7.60173782168034794738494038903, −6.85703292406759448840534925124, −6.05211650897948098090098567800, −5.46512831414042718680821100816, −4.60129753134544161179429158283, −3.89957941152044495722504266075, −2.95026025589941383464343444846, −2.52186563392001500761053665671, −1.16051571896793194894995080154, 0,
1.16051571896793194894995080154, 2.52186563392001500761053665671, 2.95026025589941383464343444846, 3.89957941152044495722504266075, 4.60129753134544161179429158283, 5.46512831414042718680821100816, 6.05211650897948098090098567800, 6.85703292406759448840534925124, 7.60173782168034794738494038903
