L(s) = 1 | − i·2-s + (0.5 + 0.866i)3-s + (0.866 − 0.5i)6-s + (0.866 − 0.5i)7-s − i·8-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s + (−0.866 + 0.5i)13-s + (−0.5 − 0.866i)14-s − 16-s + (0.866 + 0.5i)17-s + (0.866 + 0.499i)18-s + (−0.866 + 0.5i)19-s + (0.866 + 0.499i)21-s + (−0.866 + 0.5i)22-s + ⋯ |
L(s) = 1 | − i·2-s + (0.5 + 0.866i)3-s + (0.866 − 0.5i)6-s + (0.866 − 0.5i)7-s − i·8-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s + (−0.866 + 0.5i)13-s + (−0.5 − 0.866i)14-s − 16-s + (0.866 + 0.5i)17-s + (0.866 + 0.499i)18-s + (−0.866 + 0.5i)19-s + (0.866 + 0.499i)21-s + (−0.866 + 0.5i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.225874055\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.225874055\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.866 + 0.5i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + iT - T^{2} \) |
| 5 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( 1 - iT - T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + iT - T^{2} \) |
| 83 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−10.54568109257295180523296777947, −10.04852554188877455906936921779, −9.071967513984199956748669926406, −8.116150125991887498692709413491, −7.34263931032346802991061976180, −5.87911316039817196044417703740, −4.72393849642230580818059367209, −3.85123373120137279118452618740, −2.90397894227530734466906787653, −1.74362973296127457793905635046,
1.97573628137290425263419317540, 2.80999549531696596539562086700, 4.79309000686873879846392508890, 5.48219583505896501398715807099, 6.64865596757856175355698790986, 7.28691689833962311286164118778, 8.060739623734435587848126874244, 8.531254437827466964742737693384, 9.703091221328674054064600478230, 10.80807076072477096046568899450