L(s) = 1 | + (−0.866 − 0.5i)3-s + i·5-s − i·7-s + (0.499 + 0.866i)9-s − 11-s − 1.73·13-s + (0.5 − 0.866i)15-s − i·17-s + (−0.5 + 0.866i)21-s − 25-s − 0.999i·27-s − 1.73i·29-s + (0.866 + 0.5i)33-s + 35-s + (1.49 + 0.866i)39-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)3-s + i·5-s − i·7-s + (0.499 + 0.866i)9-s − 11-s − 1.73·13-s + (0.5 − 0.866i)15-s − i·17-s + (−0.5 + 0.866i)21-s − 25-s − 0.999i·27-s − 1.73i·29-s + (0.866 + 0.5i)33-s + 35-s + (1.49 + 0.866i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2759450500\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2759450500\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + 1.73T + T^{2} \) |
| 17 | \( 1 + iT - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.73iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + 1.73T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + 2T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + 1.73iT - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - 1.73T + T^{2} \) |
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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
−9.630525234398433106608012478768, −7.87197206788914487850500623746, −7.54527316966163752238153546058, −6.93760253299501339743973329889, −6.11841742206581296925811140549, −5.08690416411274267394770612853, −4.43459636065572891161287494863, −2.99762567472172961268955978983, −2.12410374074404881861011431649, −0.22042260962690694920908119877,
1.74813273874283228670279689593, 3.03815347176741101706990309041, 4.38980477155746988146585901402, 5.17160460920898847488586842526, 5.44274076881664188763265179329, 6.49508175963059169814886375249, 7.54615149348236272974007763268, 8.425302477408888190511985331131, 9.172422217338236728612709185603, 9.887337019820227846483613637431