| L(s) = 1 | − 1.34·3-s + 0.120·5-s − 0.879·7-s − 1.18·9-s − 2.71·11-s + 5.10·13-s − 0.162·15-s − 4.46·17-s + 2.87·19-s + 1.18·21-s + 5.22·23-s − 4.98·25-s + 5.63·27-s − 5.94·29-s − 1.42·31-s + 3.66·33-s − 0.106·35-s + 10.5·37-s − 6.87·39-s − 5.38·41-s + 8.27·43-s − 0.142·45-s + 6.43·47-s − 6.22·49-s + 6.01·51-s − 12.6·53-s − 0.327·55-s + ⋯ |
| L(s) = 1 | − 0.777·3-s + 0.0539·5-s − 0.332·7-s − 0.394·9-s − 0.819·11-s + 1.41·13-s − 0.0419·15-s − 1.08·17-s + 0.660·19-s + 0.258·21-s + 1.08·23-s − 0.997·25-s + 1.08·27-s − 1.10·29-s − 0.256·31-s + 0.637·33-s − 0.0179·35-s + 1.73·37-s − 1.10·39-s − 0.841·41-s + 1.26·43-s − 0.0213·45-s + 0.938·47-s − 0.889·49-s + 0.842·51-s − 1.73·53-s − 0.0441·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 3 | \( 1 + 1.34T + 3T^{2} \) |
| 5 | \( 1 - 0.120T + 5T^{2} \) |
| 7 | \( 1 + 0.879T + 7T^{2} \) |
| 11 | \( 1 + 2.71T + 11T^{2} \) |
| 13 | \( 1 - 5.10T + 13T^{2} \) |
| 17 | \( 1 + 4.46T + 17T^{2} \) |
| 19 | \( 1 - 2.87T + 19T^{2} \) |
| 23 | \( 1 - 5.22T + 23T^{2} \) |
| 29 | \( 1 + 5.94T + 29T^{2} \) |
| 31 | \( 1 + 1.42T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 5.38T + 41T^{2} \) |
| 43 | \( 1 - 8.27T + 43T^{2} \) |
| 47 | \( 1 - 6.43T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 - 2.71T + 59T^{2} \) |
| 61 | \( 1 - 1.98T + 61T^{2} \) |
| 67 | \( 1 + 2.87T + 67T^{2} \) |
| 71 | \( 1 + 9.49T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 - 7.72T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 2.64T + 89T^{2} \) |
| 97 | \( 1 - 1.68T + 97T^{2} \) |
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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
−7.53133096891590929159023724926, −6.51090887751278057931455740697, −6.13286560816782835370333252376, −5.47268524172975032338167527353, −4.83245469947467001230510454615, −3.89647525115946138172493858748, −3.12457063476171425906895145822, −2.25044830244290121555450851365, −1.04977843045961537374848881150, 0,
1.04977843045961537374848881150, 2.25044830244290121555450851365, 3.12457063476171425906895145822, 3.89647525115946138172493858748, 4.83245469947467001230510454615, 5.47268524172975032338167527353, 6.13286560816782835370333252376, 6.51090887751278057931455740697, 7.53133096891590929159023724926
