L(s) = 1 | − i·3-s + (1 + 2i)5-s − i·7-s − 9-s − 2·11-s + 6i·13-s + (2 − i)15-s − 4i·17-s − 6·19-s − 21-s + 8i·23-s + (−3 + 4i)25-s + i·27-s − 6·29-s + 2·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (0.447 + 0.894i)5-s − 0.377i·7-s − 0.333·9-s − 0.603·11-s + 1.66i·13-s + (0.516 − 0.258i)15-s − 0.970i·17-s − 1.37·19-s − 0.218·21-s + 1.66i·23-s + (−0.600 + 0.800i)25-s + 0.192i·27-s − 1.11·29-s + 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8901756374\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8901756374\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−9.448757242994790628898179101176, −9.034433095617039822921498366773, −7.63725665974598548433790479494, −7.32109692112122257685867019217, −6.46024747802954457681897307897, −5.79966116986227678363823251171, −4.64020074084346486807634327314, −3.60143476294605514230152633665, −2.47408740578303534096102298417, −1.67735342354335653842118722384,
0.31175079890528843742727667392, 1.98790849289310377535660918738, 3.01882639645572028465256492280, 4.22826466880037989105653819923, 4.98026452858904035159920529753, 5.76530168964035672627355246907, 6.36813478017749726723996899578, 7.895034858739627332379328569961, 8.396241310581547500698757024770, 8.956316066605758678104660264142