| L(s) = 1 | + i·3-s − 0.473i·5-s + 4.55·7-s − 9-s − 3.49i·11-s + 0.0840i·13-s + 0.473·15-s − 3.61·17-s + 3.61i·19-s + 4.55i·21-s + 2.82·23-s + 4.77·25-s − i·27-s − 7.30i·29-s + 0.557·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.211i·5-s + 1.72·7-s − 0.333·9-s − 1.05i·11-s + 0.0233i·13-s + 0.122·15-s − 0.877·17-s + 0.829i·19-s + 0.994i·21-s + 0.589·23-s + 0.955·25-s − 0.192i·27-s − 1.35i·29-s + 0.100·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.297251882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.297251882\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| good | 5 | \( 1 + 0.473iT - 5T^{2} \) |
| 7 | \( 1 - 4.55T + 7T^{2} \) |
| 11 | \( 1 + 3.49iT - 11T^{2} \) |
| 13 | \( 1 - 0.0840iT - 13T^{2} \) |
| 17 | \( 1 + 3.61T + 17T^{2} \) |
| 19 | \( 1 - 3.61iT - 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 7.30iT - 29T^{2} \) |
| 31 | \( 1 - 0.557T + 31T^{2} \) |
| 37 | \( 1 - 6.20iT - 37T^{2} \) |
| 41 | \( 1 - 9.27T + 41T^{2} \) |
| 43 | \( 1 + 2.27iT - 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 0.697iT - 53T^{2} \) |
| 59 | \( 1 + 5.65iT - 59T^{2} \) |
| 61 | \( 1 - 3.85iT - 61T^{2} \) |
| 67 | \( 1 + 5.33iT - 67T^{2} \) |
| 71 | \( 1 + 9.11T + 71T^{2} \) |
| 73 | \( 1 - 0.541T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + 15.0iT - 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 - 4.31T + 97T^{2} \) |
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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
−8.647457468650306945372406607523, −8.143898942675981450937704247464, −7.42155446623860053739879078338, −6.24157581805382971163154995111, −5.56527200052313555070070867727, −4.69769348402986749225301107605, −4.26333981567784134608454214335, −3.11668907299259324402401089137, −2.03853691976575106058864825599, −0.882159519218174472185509277889,
1.07635836477047447887714846415, 2.01163853133966398416553318626, 2.77001940787435140977902300231, 4.25940141473507392391287411369, 4.81717281494363520362722700581, 5.53215514210702330287444832247, 6.72049474834738839738633777570, 7.24869988463717854405666960586, 7.79331546783990371517965068567, 8.807984916115574670839425592364
