L(s) = 1 | + i·2-s + (−0.866 + 0.5i)3-s − 4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (0.866 − 0.5i)7-s − i·8-s + (−0.866 − 0.5i)10-s + (0.866 + 0.5i)11-s + (0.866 − 0.5i)12-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s − 0.999i·15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯ |
L(s) = 1 | + i·2-s + (−0.866 + 0.5i)3-s − 4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (0.866 − 0.5i)7-s − i·8-s + (−0.866 − 0.5i)10-s + (0.866 + 0.5i)11-s + (0.866 − 0.5i)12-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s − 0.999i·15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4496221474\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4496221474\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 31 | \( 1 + iT \) |
good | 3 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−14.38064540906110877033278196375, −13.18153866567090037113711101746, −11.67379482032757640573734427078, −10.90414473034916737584421224431, −9.935290020569122213739987438421, −8.387361065421477009127377174460, −7.35518884853854989998619678137, −6.30474776983992920335379169545, −5.01178891049608934019955746078, −3.94693333798037946909518849036,
1.52908883261400833847866302720, 4.01588990366557451481704433847, 5.15770407288810986410585864163, 6.51577319627012882774980448639, 8.622440126044797827118706350253, 8.768357939163126364910945396664, 10.69380714326108650488253231074, 11.61865458042174095040592607421, 11.99553861310626440190086689809, 12.91837495169102241744881554199