L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 2.23i·5-s + 1.16·7-s + 2.82i·8-s + 3.16·10-s − 5.88i·11-s + 7.16·13-s − 1.64i·14-s + 4.00·16-s + 6.32·19-s − 4.47i·20-s − 8.32·22-s + 4.47i·23-s − 5.00·25-s − 10.1i·26-s + ⋯ |
L(s) = 1 | − 0.999i·2-s − 1.00·4-s + 0.999i·5-s + 0.439·7-s + 1.00i·8-s + 1.00·10-s − 1.77i·11-s + 1.98·13-s − 0.439i·14-s + 1.00·16-s + 1.45·19-s − 1.00i·20-s − 1.77·22-s + 0.932i·23-s − 1.00·25-s − 1.98i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.20129 - 0.621838i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20129 - 0.621838i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
good | 7 | \( 1 - 1.16T + 7T^{2} \) |
| 11 | \( 1 + 5.88iT - 11T^{2} \) |
| 13 | \( 1 - 7.16T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 6.32T + 19T^{2} \) |
| 23 | \( 1 - 4.47iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 4.83T + 37T^{2} \) |
| 41 | \( 1 - 7.53iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 - 5.65iT - 53T^{2} \) |
| 59 | \( 1 + 14.3iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 0.955iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−11.22834739878756709437209641795, −10.77572798978005727751908412946, −9.629709061325891906310094833756, −8.598286009848347184001098499254, −7.85783261441437572162572179951, −6.26264692259677652491296357441, −5.41382295049435944719099842825, −3.63710201170448765273606466771, −3.17269742691337179327917510861, −1.31346534934583332374835228493,
1.36474661679798379689652479566, 3.89100685333559746365301111849, 4.80588587888937454904143072103, 5.68511512222194598575690538820, 6.88331381211836733152531048145, 7.85652118160776739448918885789, 8.682357014128935249360882932533, 9.424047137569716231696992770556, 10.42701899616266383649391610364, 11.82371335409529944941038531603