L(s) = 1 | + (1 + i)2-s + (−0.965 − 0.258i)3-s + i·4-s + (0.965 + 0.258i)5-s + (−0.707 − 1.22i)6-s + (0.866 + 0.499i)9-s + (0.707 + 1.22i)10-s + (0.5 − 0.866i)11-s + (0.258 − 0.965i)12-s + (−0.258 − 0.965i)13-s + (−0.866 − 0.499i)15-s + 16-s + (0.258 − 0.965i)17-s + (0.366 + 1.36i)18-s + (−0.258 + 0.965i)20-s + ⋯ |
L(s) = 1 | + (1 + i)2-s + (−0.965 − 0.258i)3-s + i·4-s + (0.965 + 0.258i)5-s + (−0.707 − 1.22i)6-s + (0.866 + 0.499i)9-s + (0.707 + 1.22i)10-s + (0.5 − 0.866i)11-s + (0.258 − 0.965i)12-s + (−0.258 − 0.965i)13-s + (−0.866 − 0.499i)15-s + 16-s + (0.258 − 0.965i)17-s + (0.366 + 1.36i)18-s + (−0.258 + 0.965i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.835424234\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.835424234\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.965 + 0.258i)T \) |
| 5 | \( 1 + (-0.965 - 0.258i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1 - i)T + iT^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 53 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 - 1.41iT - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (1 + i)T + iT^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 - iT - T^{2} \) |
| 83 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
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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
−9.381601251395253617841443522048, −8.198176206532694131299684868095, −7.40493194823298204128589871952, −6.64674208455232083729941800512, −6.09477128201828613346567756843, −5.50136808224775155380848334699, −4.96960490123128407764820297205, −3.90233052221617673249760603785, −2.76113752322985671704048455525, −1.22069793938991758964957970321,
1.64664842308658229977679536253, 2.05154461334820391623772397474, 3.70699514942625116558255910889, 4.22922172053155203452116963477, 5.10231697734268096186051792264, 5.68698124698926929151473915262, 6.49214448047556063450500917969, 7.30773496840162684971847667364, 8.661235633436585184371697521026, 9.556099229942269262814526423913