L(s) = 1 | + (−1.41 − i)3-s + (2.41 + 2.41i)5-s + 7-s + (1.00 + 2.82i)9-s + (1.82 − 1.82i)11-s + (−1.58 − 1.58i)13-s + (−1 − 5.82i)15-s − 6.82i·17-s + (2.41 − 2.41i)19-s + (−1.41 − i)21-s + 3.65i·23-s + 6.65i·25-s + (1.41 − 5.00i)27-s + (3 − 3i)29-s + 10.4i·31-s + ⋯ |
L(s) = 1 | + (−0.816 − 0.577i)3-s + (1.07 + 1.07i)5-s + 0.377·7-s + (0.333 + 0.942i)9-s + (0.551 − 0.551i)11-s + (−0.439 − 0.439i)13-s + (−0.258 − 1.50i)15-s − 1.65i·17-s + (0.553 − 0.553i)19-s + (−0.308 − 0.218i)21-s + 0.762i·23-s + 1.33i·25-s + (0.272 − 0.962i)27-s + (0.557 − 0.557i)29-s + 1.88i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.680012956\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.680012956\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.41 + i)T \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + (-2.41 - 2.41i)T + 5iT^{2} \) |
| 11 | \( 1 + (-1.82 + 1.82i)T - 11iT^{2} \) |
| 13 | \( 1 + (1.58 + 1.58i)T + 13iT^{2} \) |
| 17 | \( 1 + 6.82iT - 17T^{2} \) |
| 19 | \( 1 + (-2.41 + 2.41i)T - 19iT^{2} \) |
| 23 | \( 1 - 3.65iT - 23T^{2} \) |
| 29 | \( 1 + (-3 + 3i)T - 29iT^{2} \) |
| 31 | \( 1 - 10.4iT - 31T^{2} \) |
| 37 | \( 1 + (3.82 - 3.82i)T - 37iT^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + (1.82 + 1.82i)T + 43iT^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 + (-0.171 - 0.171i)T + 53iT^{2} \) |
| 59 | \( 1 + (-4.07 + 4.07i)T - 59iT^{2} \) |
| 61 | \( 1 + (0.414 + 0.414i)T + 61iT^{2} \) |
| 67 | \( 1 + (-7 + 7i)T - 67iT^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 - 10.8iT - 73T^{2} \) |
| 79 | \( 1 + 2iT - 79T^{2} \) |
| 83 | \( 1 + (10.8 + 10.8i)T + 83iT^{2} \) |
| 89 | \( 1 - 4.34T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−9.775431799804564253709374446117, −8.886707899886316837341324790307, −7.59773924285304624014419332543, −7.03375832885686614286152040934, −6.34872863457393481072341001023, −5.48545835291658278167263262439, −4.87246908692319516689750274514, −3.15897096146462715546894523367, −2.31050775647320779079563945715, −0.999885880031456449031433174620,
1.10244403189993113319304630940, 2.10912983494208314494426101979, 4.04067543379777126071146160096, 4.52199898647944002307840861968, 5.59348558860047439403756038799, 5.97588474132152370419127524711, 7.00413485143433552534941608471, 8.225047024725725113722899525621, 9.086170645893138695341390865773, 9.651195316338329585832305235775