| L(s) = 1 | + (−2.20 + 0.349i)5-s + 2.26i·7-s − 2.54·11-s − 5.02i·13-s − 5.72i·17-s + 1.86·19-s + 5.39i·23-s + (4.75 − 1.54i)25-s + 3·29-s + 9.38·31-s + (−0.791 − 5.00i)35-s + 2.59i·37-s + 3.96·41-s + 10.2i·43-s − 1.07i·47-s + ⋯ |
| L(s) = 1 | + (−0.987 + 0.156i)5-s + 0.856i·7-s − 0.768·11-s − 1.39i·13-s − 1.38i·17-s + 0.429·19-s + 1.12i·23-s + (0.951 − 0.308i)25-s + 0.557·29-s + 1.68·31-s + (−0.133 − 0.845i)35-s + 0.426i·37-s + 0.619·41-s + 1.55i·43-s − 0.156i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.255390723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.255390723\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.20 - 0.349i)T \) |
| good | 7 | \( 1 - 2.26iT - 7T^{2} \) |
| 11 | \( 1 + 2.54T + 11T^{2} \) |
| 13 | \( 1 + 5.02iT - 13T^{2} \) |
| 17 | \( 1 + 5.72iT - 17T^{2} \) |
| 19 | \( 1 - 1.86T + 19T^{2} \) |
| 23 | \( 1 - 5.39iT - 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 9.38T + 31T^{2} \) |
| 37 | \( 1 - 2.59iT - 37T^{2} \) |
| 41 | \( 1 - 3.96T + 41T^{2} \) |
| 43 | \( 1 - 10.2iT - 43T^{2} \) |
| 47 | \( 1 + 1.07iT - 47T^{2} \) |
| 53 | \( 1 - 10.7iT - 53T^{2} \) |
| 59 | \( 1 - 1.58T + 59T^{2} \) |
| 61 | \( 1 - 0.869T + 61T^{2} \) |
| 67 | \( 1 - 4.85iT - 67T^{2} \) |
| 71 | \( 1 + 1.86T + 71T^{2} \) |
| 73 | \( 1 - 3.87iT - 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 + 14.7iT - 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 17.6iT - 97T^{2} \) |
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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
−9.418817167734446813862059370726, −8.443985995722925767512584847798, −7.83407810643897455311194021226, −7.26858859559232151309451822275, −6.07594495266566320041881651826, −5.25417250894017988459131177892, −4.53815840055412135535226853737, −3.06601709165197626600027130063, −2.79038081384094175277613527694, −0.78083858101985060188693368648,
0.77553547349887033334375542439, 2.28028821726144664956568555637, 3.63866500471855697411548361755, 4.25648610684150837396868777163, 5.01920239658685234725158719656, 6.38326631926305902110268505508, 6.96224608112415382239138651822, 7.913389601894629882159708930447, 8.382770438115062917036711913117, 9.294785719758523729189416115555
