L(s) = 1 | + (0.866 − 0.5i)2-s + (0.707 − 0.707i)3-s + (0.499 − 0.866i)4-s + (−0.707 − 0.707i)5-s + (0.258 − 0.965i)6-s + (−0.866 + 0.5i)7-s − 0.999i·8-s − 1.00i·9-s + (−0.965 − 0.258i)10-s + (−0.258 − 0.965i)12-s + (−1.22 − 0.707i)13-s + (−0.499 + 0.866i)14-s − 1.00·15-s + (−0.5 − 0.866i)16-s + (−0.500 − 0.866i)18-s + 0.517·19-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (0.707 − 0.707i)3-s + (0.499 − 0.866i)4-s + (−0.707 − 0.707i)5-s + (0.258 − 0.965i)6-s + (−0.866 + 0.5i)7-s − 0.999i·8-s − 1.00i·9-s + (−0.965 − 0.258i)10-s + (−0.258 − 0.965i)12-s + (−1.22 − 0.707i)13-s + (−0.499 + 0.866i)14-s − 1.00·15-s + (−0.5 − 0.866i)16-s + (−0.500 − 0.866i)18-s + 0.517·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.841095004\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.841095004\) |
\(L(1)\) |
|
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 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
good | 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - 0.517T + T^{2} \) |
| 23 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 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 | \( 1 + T^{2} \) |
| 59 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - 1.73T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 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
−8.948745407946266898792103355178, −7.81229737795408855359043248703, −7.26035960844685839390702844381, −6.47344478007787475006545129792, −5.43631966549747096674955100258, −4.85144894246307206655767678446, −3.56708211409875835378910969579, −3.13743103027962582729479615872, −2.16520029951321717550769373687, −0.820003469803503444506877484270,
2.50156684372160513870310925946, 3.02364164082644446447693374005, 3.88415936102061332209783359621, 4.49593135173543455701083598967, 5.31308738602384513798855200464, 6.58297695429233348371553541810, 7.10478915032948939157870407181, 7.63345502476727761524521801436, 8.562370949862687615197306255382, 9.362195040193264947020182501427