L(s) = 1 | + i·3-s + i·7-s − 9-s − 2·13-s + 2i·19-s − 21-s + 25-s − i·27-s − 2i·39-s − 49-s − 2·57-s + 2·61-s − i·63-s + i·75-s + 2i·79-s + ⋯ |
L(s) = 1 | + i·3-s + i·7-s − 9-s − 2·13-s + 2i·19-s − 21-s + 25-s − i·27-s − 2i·39-s − 49-s − 2·57-s + 2·61-s − i·63-s + i·75-s + 2i·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8157877663\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8157877663\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 2T + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - 2iT - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - 2T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 2iT - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.902396192078867387683212222100, −9.545000781749769850760113908114, −8.565175954157963719945820308180, −7.916189395212009002813251088120, −6.77837805955879714663078744850, −5.64798320921760840170288700174, −5.16939002276792563186168539567, −4.20600723611712189144672787205, −3.08569452244465344015797761851, −2.19120588441234514115597611919,
0.65597087790900529738681733472, 2.23105734622805977135489512021, 3.08120214068228317677806424379, 4.57784368867396479056914895631, 5.20353217517582207709577092423, 6.57273757678871610247577869121, 7.13227714042248432464393531804, 7.57730701215358861084635402820, 8.633981357776708861677121797666, 9.457818700670686484362034482537