| L(s) = 1 | − 0.537·3-s + 5-s − 1.46·7-s − 2.71·9-s + 5.03·11-s − 0.136·13-s − 0.537·15-s − 3.68·17-s + 4.39·19-s + 0.785·21-s − 4.79·23-s + 25-s + 3.07·27-s + 8.69·29-s − 7.84·31-s − 2.70·33-s − 1.46·35-s − 9.40·37-s + 0.0734·39-s + 1.40·41-s − 10.2·43-s − 2.71·45-s + 9.20·47-s − 4.86·49-s + 1.98·51-s + 9.68·53-s + 5.03·55-s + ⋯ |
| L(s) = 1 | − 0.310·3-s + 0.447·5-s − 0.552·7-s − 0.903·9-s + 1.51·11-s − 0.0378·13-s − 0.138·15-s − 0.893·17-s + 1.00·19-s + 0.171·21-s − 0.998·23-s + 0.200·25-s + 0.591·27-s + 1.61·29-s − 1.40·31-s − 0.471·33-s − 0.246·35-s − 1.54·37-s + 0.0117·39-s + 0.219·41-s − 1.55·43-s − 0.404·45-s + 1.34·47-s − 0.695·49-s + 0.277·51-s + 1.32·53-s + 0.679·55-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 | \( 1 + 0.537T + 3T^{2} \) |
| 7 | \( 1 + 1.46T + 7T^{2} \) |
| 11 | \( 1 - 5.03T + 11T^{2} \) |
| 13 | \( 1 + 0.136T + 13T^{2} \) |
| 17 | \( 1 + 3.68T + 17T^{2} \) |
| 19 | \( 1 - 4.39T + 19T^{2} \) |
| 23 | \( 1 + 4.79T + 23T^{2} \) |
| 29 | \( 1 - 8.69T + 29T^{2} \) |
| 31 | \( 1 + 7.84T + 31T^{2} \) |
| 37 | \( 1 + 9.40T + 37T^{2} \) |
| 41 | \( 1 - 1.40T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 9.20T + 47T^{2} \) |
| 53 | \( 1 - 9.68T + 53T^{2} \) |
| 59 | \( 1 - 2.31T + 59T^{2} \) |
| 61 | \( 1 - 8.56T + 61T^{2} \) |
| 67 | \( 1 + 9.51T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 4.59T + 73T^{2} \) |
| 79 | \( 1 - 8.44T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 1.67T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−7.18997421376411580743609557054, −6.77957619169347526134197796776, −6.07457626687319712220292275968, −5.57663127520863316476963385500, −4.72085956068183399502343622222, −3.82474363447936874952863352156, −3.16895150008651479847257020283, −2.20617788436427318996774172718, −1.23825042124803306563903760475, 0,
1.23825042124803306563903760475, 2.20617788436427318996774172718, 3.16895150008651479847257020283, 3.82474363447936874952863352156, 4.72085956068183399502343622222, 5.57663127520863316476963385500, 6.07457626687319712220292275968, 6.77957619169347526134197796776, 7.18997421376411580743609557054
