| L(s) = 1 | + 1.27·3-s + 2.87·5-s + 7-s − 1.38·9-s + 1.27·11-s + 13-s + 3.65·15-s + 3.10·17-s − 0.870·19-s + 1.27·21-s − 0.402·23-s + 3.23·25-s − 5.57·27-s + 6.52·29-s − 5.79·31-s + 1.61·33-s + 2.87·35-s + 0.727·37-s + 1.27·39-s − 6.12·41-s − 0.237·43-s − 3.96·45-s − 5.79·47-s + 49-s + 3.95·51-s − 2.52·53-s + 3.65·55-s + ⋯ |
| L(s) = 1 | + 0.734·3-s + 1.28·5-s + 0.377·7-s − 0.460·9-s + 0.383·11-s + 0.277·13-s + 0.943·15-s + 0.753·17-s − 0.199·19-s + 0.277·21-s − 0.0839·23-s + 0.647·25-s − 1.07·27-s + 1.21·29-s − 1.04·31-s + 0.281·33-s + 0.485·35-s + 0.119·37-s + 0.203·39-s − 0.955·41-s − 0.0361·43-s − 0.590·45-s − 0.845·47-s + 0.142·49-s + 0.553·51-s − 0.346·53-s + 0.492·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.430117185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.430117185\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.27T + 3T^{2} \) |
| 5 | \( 1 - 2.87T + 5T^{2} \) |
| 11 | \( 1 - 1.27T + 11T^{2} \) |
| 17 | \( 1 - 3.10T + 17T^{2} \) |
| 19 | \( 1 + 0.870T + 19T^{2} \) |
| 23 | \( 1 + 0.402T + 23T^{2} \) |
| 29 | \( 1 - 6.52T + 29T^{2} \) |
| 31 | \( 1 + 5.79T + 31T^{2} \) |
| 37 | \( 1 - 0.727T + 37T^{2} \) |
| 41 | \( 1 + 6.12T + 41T^{2} \) |
| 43 | \( 1 + 0.237T + 43T^{2} \) |
| 47 | \( 1 + 5.79T + 47T^{2} \) |
| 53 | \( 1 + 2.52T + 53T^{2} \) |
| 59 | \( 1 - 3.19T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 4.92T + 67T^{2} \) |
| 71 | \( 1 + 1.43T + 71T^{2} \) |
| 73 | \( 1 - 3.51T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + 6.32T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 + 6.34T + 97T^{2} \) |
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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
−10.16202754756115469374544348173, −9.533017541097882487239802969546, −8.695492418615804109979979885270, −8.056159228484255979045690684334, −6.83130072223761712259740650855, −5.90067290745228996105072215525, −5.12358655378588464258905609593, −3.69790577136341211603372880299, −2.59505455189682805055359257629, −1.55452007153238320198007779500,
1.55452007153238320198007779500, 2.59505455189682805055359257629, 3.69790577136341211603372880299, 5.12358655378588464258905609593, 5.90067290745228996105072215525, 6.83130072223761712259740650855, 8.056159228484255979045690684334, 8.695492418615804109979979885270, 9.533017541097882487239802969546, 10.16202754756115469374544348173
