L(s) = 1 | + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s + 17-s − 19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 + 0.866i)29-s + (−0.5 + 0.866i)31-s − 35-s − 37-s + (−0.5 + 0.866i)41-s + (0.5 + 0.866i)43-s + (−0.5 − 0.866i)47-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s + 17-s − 19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 + 0.866i)29-s + (−0.5 + 0.866i)31-s − 35-s − 37-s + (−0.5 + 0.866i)41-s + (0.5 + 0.866i)43-s + (−0.5 − 0.866i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9197214877 - 0.3347512453i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9197214877 - 0.3347512453i\) |
\(L(1)\) |
\(\approx\) |
\(1.035205736 - 0.2134755561i\) |
\(L(1)\) |
\(\approx\) |
\(1.035205736 - 0.2134755561i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + (0.5 - 0.866i)T \) |
| 61 | \( 1 + (0.5 + 0.866i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−31.81345037823403048832936559512, −30.557319028048212582972783100226, −29.619248052843133548409247498817, −28.6139255685274555331863775509, −27.41517636242089709720143388095, −26.10979805331162340313045100797, −25.44082151383310473372851304468, −24.17102377076190764350303422985, −22.780363239488029427159042612252, −21.85934278670287855152163372819, −21.01379150198451652054836439490, −18.98622330756815119322015081036, −18.81551373666136698933800628125, −17.2041748756147054666435420123, −16.00173978074129690032718703065, −14.67400468122114456720768227004, −13.73846673582274412699715473714, −12.23350348113595704231288338606, −11.007121727582652084083529766350, −9.69775569278157298222034893172, −8.491082575023549961928390227699, −6.6234689328207247566273298344, −5.82255136630110305200079307601, −3.669186280593918975584787366677, −2.22597710360548271714684502500,
1.42467481873239634771439473912, 3.629394625628101399085476972088, 5.10629851236419492999352337039, 6.57991586841707826170401745110, 8.08358848300956800406786806195, 9.52242381613333263134807422902, 10.484692531373141296415547819509, 12.3133561253029776154034714594, 13.1595509650851485054737479288, 14.39184439348052417964155640728, 15.937349668853705008737854827317, 16.983241051665548410315856266625, 17.85631056047438788033707824402, 19.60071893007075376862556111168, 20.36519722341295997147191642723, 21.46661557659550218001629314767, 22.9122346915152802393354639874, 23.72435340757309339707721143610, 25.26132519103447050704413751036, 25.72186739323419272356986733161, 27.42542855836235172091111798517, 28.163561540372382389635507260242, 29.48421580941850672573034809243, 30.17375131833142524894927598414, 31.71905557787174441538596254243