L(s) = 1 | + (0.5 + 0.866i)2-s + (0.5 + 0.866i)5-s + 8-s + (−0.5 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (0.5 + 0.866i)16-s + (0.499 − 0.866i)18-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)31-s + (−0.499 + 0.866i)38-s + (0.499 + 0.866i)40-s − 41-s + (0.499 − 0.866i)45-s + (−1 − 1.73i)47-s + (0.5 − 0.866i)59-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (0.5 + 0.866i)5-s + 8-s + (−0.5 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (0.5 + 0.866i)16-s + (0.499 − 0.866i)18-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)31-s + (−0.499 + 0.866i)38-s + (0.499 + 0.866i)40-s − 41-s + (0.499 − 0.866i)45-s + (−1 − 1.73i)47-s + (0.5 − 0.866i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.676892460\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.676892460\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 3 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−10.02615308589040029173012744557, −8.902264164128842793892442054404, −8.036722901853991563032676656790, −7.00041372742349385275271944046, −6.61150482595742757980419471094, −5.80131568641837758434104961341, −5.18874105438898270256957796624, −3.91456708499095300231461329588, −3.00813035561280530498233893967, −1.66360725819855976484094091015,
1.44245599534422989729290570143, 2.42332882596807080427336740261, 3.33053878687794730611647045656, 4.57702365163549852515790679274, 5.04539794613731082652622753114, 5.97700974673574873453828731241, 7.24490233020250215802836124123, 7.952905096785869610920164318591, 8.833577430373471413404445248365, 9.587564340529044906171767578752