| L(s) = 1 | − 1.33·2-s + 0.118·3-s − 0.205·4-s + 0.202·5-s − 0.158·6-s + 4.70·7-s + 2.95·8-s − 2.98·9-s − 0.271·10-s + 2.81·11-s − 0.0242·12-s − 6.30·14-s + 0.0239·15-s − 3.54·16-s − 1.69·17-s + 4.00·18-s − 1.56·19-s − 0.0415·20-s + 0.557·21-s − 3.77·22-s + 1.98·23-s + 0.349·24-s − 4.95·25-s − 0.708·27-s − 0.966·28-s + 7.44·29-s − 0.0320·30-s + ⋯ |
| L(s) = 1 | − 0.947·2-s + 0.0682·3-s − 0.102·4-s + 0.0905·5-s − 0.0646·6-s + 1.77·7-s + 1.04·8-s − 0.995·9-s − 0.0857·10-s + 0.849·11-s − 0.00700·12-s − 1.68·14-s + 0.00618·15-s − 0.886·16-s − 0.411·17-s + 0.942·18-s − 0.358·19-s − 0.00928·20-s + 0.121·21-s − 0.804·22-s + 0.412·23-s + 0.0713·24-s − 0.991·25-s − 0.136·27-s − 0.182·28-s + 1.38·29-s − 0.00585·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.327420929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.327420929\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.33T + 2T^{2} \) |
| 3 | \( 1 - 0.118T + 3T^{2} \) |
| 5 | \( 1 - 0.202T + 5T^{2} \) |
| 7 | \( 1 - 4.70T + 7T^{2} \) |
| 11 | \( 1 - 2.81T + 11T^{2} \) |
| 17 | \( 1 + 1.69T + 17T^{2} \) |
| 19 | \( 1 + 1.56T + 19T^{2} \) |
| 23 | \( 1 - 1.98T + 23T^{2} \) |
| 29 | \( 1 - 7.44T + 29T^{2} \) |
| 37 | \( 1 - 8.73T + 37T^{2} \) |
| 41 | \( 1 + 1.99T + 41T^{2} \) |
| 43 | \( 1 + 1.07T + 43T^{2} \) |
| 47 | \( 1 + 0.861T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 5.66T + 67T^{2} \) |
| 71 | \( 1 + 2.34T + 71T^{2} \) |
| 73 | \( 1 + 1.28T + 73T^{2} \) |
| 79 | \( 1 + 7.62T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−8.362478348569677502417269780047, −7.84667554600296277450581673387, −6.99520492520390274133277996674, −6.10061786438106003475965566440, −5.17953047119995190817831049077, −4.58649627957046436569901239444, −3.86428203970324584951108842029, −2.47497870173357723047323833250, −1.67149946429003813229916344086, −0.76359759092130914832435078099,
0.76359759092130914832435078099, 1.67149946429003813229916344086, 2.47497870173357723047323833250, 3.86428203970324584951108842029, 4.58649627957046436569901239444, 5.17953047119995190817831049077, 6.10061786438106003475965566440, 6.99520492520390274133277996674, 7.84667554600296277450581673387, 8.362478348569677502417269780047
