| L(s) = 1 | + 6.32·3-s − 17.8·5-s + 22.6·7-s + 13.0·9-s + 44.2·11-s − 17.8·13-s − 113.·15-s + 70·17-s + 82.2·19-s + 143.·21-s + 158.·23-s + 195.·25-s − 88.5·27-s + 125.·29-s + 280·33-s − 404.·35-s − 375.·37-s − 113.·39-s + 182·41-s − 132.·43-s − 232.·45-s + 316.·47-s + 169.·49-s + 442.·51-s + 125.·53-s − 791.·55-s + 520·57-s + ⋯ |
| L(s) = 1 | + 1.21·3-s − 1.60·5-s + 1.22·7-s + 0.481·9-s + 1.21·11-s − 0.381·13-s − 1.94·15-s + 0.998·17-s + 0.992·19-s + 1.48·21-s + 1.43·23-s + 1.56·25-s − 0.631·27-s + 0.801·29-s + 1.47·33-s − 1.95·35-s − 1.66·37-s − 0.464·39-s + 0.693·41-s − 0.471·43-s − 0.770·45-s + 0.983·47-s + 0.492·49-s + 1.21·51-s + 0.324·53-s − 1.94·55-s + 1.20·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.526310059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.526310059\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 6.32T + 27T^{2} \) |
| 5 | \( 1 + 17.8T + 125T^{2} \) |
| 7 | \( 1 - 22.6T + 343T^{2} \) |
| 11 | \( 1 - 44.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 17.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 70T + 4.91e3T^{2} \) |
| 19 | \( 1 - 82.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 158.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 125.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 2.97e4T^{2} \) |
| 37 | \( 1 + 375.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 182T + 6.89e4T^{2} \) |
| 43 | \( 1 + 132.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 316.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 125.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 82.2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 232.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 221.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 113.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 910T + 3.89e5T^{2} \) |
| 79 | \( 1 - 678.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 714.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 546T + 7.04e5T^{2} \) |
| 97 | \( 1 + 490T + 9.12e5T^{2} \) |
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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
−11.79241142976214084938224259015, −10.78053927458870975516483751194, −9.285986175938985519340728425500, −8.532196739042168954869126170542, −7.77110021882884065442369727389, −7.12573394853413171373022686224, −5.06133754337830436455469454741, −3.94581939992963004162318102018, −3.06148260417604879894659198697, −1.22451930226900963150378655118,
1.22451930226900963150378655118, 3.06148260417604879894659198697, 3.94581939992963004162318102018, 5.06133754337830436455469454741, 7.12573394853413171373022686224, 7.77110021882884065442369727389, 8.532196739042168954869126170542, 9.285986175938985519340728425500, 10.78053927458870975516483751194, 11.79241142976214084938224259015
