| L(s) = 1 | − 5·5-s + 18.4·7-s − 55.3·11-s + 52·13-s − 66·17-s − 36.8·23-s + 25·25-s + 46·29-s + 258.·31-s − 92.1·35-s − 24·37-s − 72·41-s + 331.·47-s − 3·49-s + 62·53-s + 276.·55-s + 239.·59-s − 190·61-s − 260·65-s − 774.·67-s + 626.·71-s + 1.07e3·73-s − 1.02e3·77-s − 848.·79-s − 774.·83-s + 330·85-s + 1.11e3·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.995·7-s − 1.51·11-s + 1.10·13-s − 0.941·17-s − 0.334·23-s + 0.200·25-s + 0.294·29-s + 1.49·31-s − 0.445·35-s − 0.106·37-s − 0.274·41-s + 1.03·47-s − 0.00874·49-s + 0.160·53-s + 0.678·55-s + 0.528·59-s − 0.398·61-s − 0.496·65-s − 1.41·67-s + 1.04·71-s + 1.72·73-s − 1.50·77-s − 1.20·79-s − 1.02·83-s + 0.421·85-s + 1.32·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.882690156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.882690156\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 - 18.4T + 343T^{2} \) |
| 11 | \( 1 + 55.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 52T + 2.19e3T^{2} \) |
| 17 | \( 1 + 66T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 + 36.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 46T + 2.43e4T^{2} \) |
| 31 | \( 1 - 258.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 24T + 5.06e4T^{2} \) |
| 41 | \( 1 + 72T + 6.89e4T^{2} \) |
| 43 | \( 1 + 7.95e4T^{2} \) |
| 47 | \( 1 - 331.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 62T + 1.48e5T^{2} \) |
| 59 | \( 1 - 239.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 190T + 2.26e5T^{2} \) |
| 67 | \( 1 + 774.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 626.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.07e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 848.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 774.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.11e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 834T + 9.12e5T^{2} \) |
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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
−8.912856126298611263862799353082, −8.244334347151921423495337569556, −7.80633492950746166075816464811, −6.76981510539916783623897697761, −5.79663477401005337928127115655, −4.88811913627319441901115687963, −4.20403482585620005909207910449, −2.99282074878415759420141897701, −1.97227854770729296873064399172, −0.67387880840112219632033262411,
0.67387880840112219632033262411, 1.97227854770729296873064399172, 2.99282074878415759420141897701, 4.20403482585620005909207910449, 4.88811913627319441901115687963, 5.79663477401005337928127115655, 6.76981510539916783623897697761, 7.80633492950746166075816464811, 8.244334347151921423495337569556, 8.912856126298611263862799353082
