L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.866 − 0.5i)5-s − 0.999i·8-s + (0.499 − 0.866i)10-s + (−0.5 − 0.866i)16-s + i·17-s + 1.73i·19-s − 0.999i·20-s + (−0.866 − 1.5i)23-s + (0.499 − 0.866i)25-s + (−1.5 + 0.866i)31-s + (−0.866 − 0.499i)32-s + (0.5 + 0.866i)34-s + (0.866 + 1.49i)38-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.866 − 0.5i)5-s − 0.999i·8-s + (0.499 − 0.866i)10-s + (−0.5 − 0.866i)16-s + i·17-s + 1.73i·19-s − 0.999i·20-s + (−0.866 − 1.5i)23-s + (0.499 − 0.866i)25-s + (−1.5 + 0.866i)31-s + (−0.866 − 0.499i)32-s + (0.5 + 0.866i)34-s + (0.866 + 1.49i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.059693657\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.059693657\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
good | 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - iT - T^{2} \) |
| 19 | \( 1 - 1.73iT - T^{2} \) |
| 23 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + iT - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−9.641281582971000526324660776285, −8.766064760443898842999176615245, −7.88588828543812388200220977461, −6.62642737977617720361907892094, −5.99184138148652764590891469838, −5.36737219805671584897445819349, −4.35764931623525485525467436163, −3.57255283615605177361288474575, −2.26242527132989234558910948549, −1.48230050296191406741310029786,
2.02431055089821508885389727722, 2.88976221028055071545930440726, 3.86289823568665006911219669556, 5.04470615060909331958343355146, 5.57197238257233679096340628042, 6.49160182955838938500626798135, 7.17984087081652675624715232000, 7.80459747186206550250283137070, 9.123824670669407428140685423797, 9.488478737050878408180406797730