L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (1.22 + 2.12i)5-s + (−1 − 2.44i)7-s + 0.999i·8-s + (−2.12 − 1.22i)10-s + (−0.866 − 0.5i)11-s − 2.44i·13-s + (2.09 + 1.62i)14-s + (−0.5 − 0.866i)16-s + (0.866 − 1.5i)17-s + (−1.5 + 0.866i)19-s + 2.44·20-s + 0.999·22-s + (−1.07 + 0.621i)23-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.547 + 0.948i)5-s + (−0.377 − 0.925i)7-s + 0.353i·8-s + (−0.670 − 0.387i)10-s + (−0.261 − 0.150i)11-s − 0.679i·13-s + (0.558 + 0.433i)14-s + (−0.125 − 0.216i)16-s + (0.210 − 0.363i)17-s + (−0.344 + 0.198i)19-s + 0.547·20-s + 0.213·22-s + (−0.224 + 0.129i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.410 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9089232774\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9089232774\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1 + 2.44i)T \) |
| 11 | \( 1 + (0.866 + 0.5i)T \) |
good | 5 | \( 1 + (-1.22 - 2.12i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 + (-0.866 + 1.5i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.07 - 0.621i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 7.24iT - 29T^{2} \) |
| 31 | \( 1 + (7.24 + 4.18i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.62 + 4.54i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 8.36T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + (1.37 + 2.37i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.30 - 3.98i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-9 + 5.19i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.24 - 3.88i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 1.24iT - 71T^{2} \) |
| 73 | \( 1 + (4.24 + 2.44i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 7.94T + 83T^{2} \) |
| 89 | \( 1 + (-2.95 - 5.12i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 6.63iT - 97T^{2} \) |
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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
−9.708484705143657379454642323884, −8.585408910169913929301937714097, −7.57021684937986702427255611667, −7.21094747845942552182034397946, −6.16800769161298754043182702861, −5.66768720051519828286214799695, −4.24288101465527373997218789787, −3.15028136939763680781396950707, −2.11169663435721733875371032799, −0.45492873854361321779354912078,
1.39137989403283734565056790995, 2.30680172020370092263422997709, 3.48574367099021206298892565246, 4.76230908977145404416609053270, 5.54988868575649781454133669258, 6.45674044701306664277197051271, 7.40270039900679333102873621304, 8.548476477379778955376012152234, 8.930501448848246347442066146070, 9.526024951445909016634482736809