| L(s) = 1 | + 2-s + 4-s − 3.35i·5-s − i·7-s + 8-s − 3.35i·10-s + (−3.06 + 1.25i)11-s − 2.62i·13-s − i·14-s + 16-s − 2.58·17-s − 7.21i·19-s − 3.35i·20-s + (−3.06 + 1.25i)22-s + 5.09i·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.49i·5-s − 0.377i·7-s + 0.353·8-s − 1.06i·10-s + (−0.925 + 0.379i)11-s − 0.728i·13-s − 0.267i·14-s + 0.250·16-s − 0.627·17-s − 1.65i·19-s − 0.749i·20-s + (−0.654 + 0.268i)22-s + 1.06i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.536 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.536 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.997944796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.997944796\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (3.06 - 1.25i)T \) |
| good | 5 | \( 1 + 3.35iT - 5T^{2} \) |
| 13 | \( 1 + 2.62iT - 13T^{2} \) |
| 17 | \( 1 + 2.58T + 17T^{2} \) |
| 19 | \( 1 + 7.21iT - 19T^{2} \) |
| 23 | \( 1 - 5.09iT - 23T^{2} \) |
| 29 | \( 1 - 1.96T + 29T^{2} \) |
| 31 | \( 1 + 4.93T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 + 9.07T + 41T^{2} \) |
| 43 | \( 1 + 4.02iT - 43T^{2} \) |
| 47 | \( 1 - 7.16iT - 47T^{2} \) |
| 53 | \( 1 + 10.3iT - 53T^{2} \) |
| 59 | \( 1 + 11.5iT - 59T^{2} \) |
| 61 | \( 1 + 4.07iT - 61T^{2} \) |
| 67 | \( 1 + 2.06T + 67T^{2} \) |
| 71 | \( 1 - 11.7iT - 71T^{2} \) |
| 73 | \( 1 - 5.29iT - 73T^{2} \) |
| 79 | \( 1 - 3.51iT - 79T^{2} \) |
| 83 | \( 1 - 2.98T + 83T^{2} \) |
| 89 | \( 1 + 12.8iT - 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
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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
−9.303420181973958883526465157341, −8.421265829776933709076456546246, −7.68969874932571711078524127636, −6.84246222591047736001120903034, −5.64246652821101445241933906889, −4.97583690441240665217967384565, −4.47197445308381907869664866460, −3.25016540366559068622060001037, −2.02783044405999571758107709594, −0.60204228706621297474866295032,
2.06078779722428732377300390664, 2.83560836701747833460727565503, 3.72106653068694719512404402842, 4.75205608797531803960912504464, 5.96376370860821411508758804887, 6.33650264134124708014079475669, 7.29970256961241967789272342401, 8.000776445417007172527934365121, 9.040148460311765273526920350231, 10.32396371093469679498679724737
