L(s) = 1 | + (0.991 − 0.130i)3-s + (−0.130 + 0.991i)5-s + (0.965 − 0.258i)9-s + (0.608 − 0.793i)11-s + (0.923 + 0.382i)13-s + i·15-s + (0.866 − 0.5i)17-s + (−0.793 + 0.608i)19-s + (0.965 − 0.258i)23-s + (−0.965 − 0.258i)25-s + (0.923 − 0.382i)27-s + (−0.382 + 0.923i)29-s + (−0.5 − 0.866i)31-s + (0.5 − 0.866i)33-s + (0.130 − 0.991i)37-s + ⋯ |
L(s) = 1 | + (0.991 − 0.130i)3-s + (−0.130 + 0.991i)5-s + (0.965 − 0.258i)9-s + (0.608 − 0.793i)11-s + (0.923 + 0.382i)13-s + i·15-s + (0.866 − 0.5i)17-s + (−0.793 + 0.608i)19-s + (0.965 − 0.258i)23-s + (−0.965 − 0.258i)25-s + (0.923 − 0.382i)27-s + (−0.382 + 0.923i)29-s + (−0.5 − 0.866i)31-s + (0.5 − 0.866i)33-s + (0.130 − 0.991i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.935 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.935 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.337538375 + 0.6108631396i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.337538375 + 0.6108631396i\) |
\(L(1)\) |
\(\approx\) |
\(1.698466313 + 0.1614655844i\) |
\(L(1)\) |
\(\approx\) |
\(1.698466313 + 0.1614655844i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.991 - 0.130i)T \) |
| 5 | \( 1 + (-0.130 + 0.991i)T \) |
| 11 | \( 1 + (0.608 - 0.793i)T \) |
| 13 | \( 1 + (0.923 + 0.382i)T \) |
| 17 | \( 1 + (0.866 - 0.5i)T \) |
| 19 | \( 1 + (-0.793 + 0.608i)T \) |
| 23 | \( 1 + (0.965 - 0.258i)T \) |
| 29 | \( 1 + (-0.382 + 0.923i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.130 - 0.991i)T \) |
| 41 | \( 1 + (0.707 + 0.707i)T \) |
| 43 | \( 1 + (-0.382 - 0.923i)T \) |
| 47 | \( 1 + (0.866 + 0.5i)T \) |
| 53 | \( 1 + (-0.608 + 0.793i)T \) |
| 59 | \( 1 + (0.793 + 0.608i)T \) |
| 61 | \( 1 + (0.608 + 0.793i)T \) |
| 67 | \( 1 + (0.991 - 0.130i)T \) |
| 71 | \( 1 + (0.707 - 0.707i)T \) |
| 73 | \( 1 + (-0.258 + 0.965i)T \) |
| 79 | \( 1 + (-0.866 - 0.5i)T \) |
| 83 | \( 1 + (0.923 + 0.382i)T \) |
| 89 | \( 1 + (0.258 + 0.965i)T \) |
| 97 | \( 1 - T \) |
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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
−23.77682962109054384029069263860, −23.09251108862247803710279528103, −21.769652764049735637684638254729, −20.94029572770178397946691160565, −20.416130531770354687062819523238, −19.55359626098880456966304356814, −18.88820577960435871872286496553, −17.60099657903070154856322858043, −16.8074928052937248845466930264, −15.75241777899397876467781318229, −15.105010495302691867047820867142, −14.17691432749836209898149641134, −13.04697082168894270476628804159, −12.69660647313504100666217303327, −11.415882577607481944919921295969, −10.17164418866105924615891645881, −9.28415846348025248638044880115, −8.56330915455747546694410283149, −7.77996837674679072115453288477, −6.62994458419084165850772577660, −5.236815244539506974037063570063, −4.23491276729254010767516585727, −3.41855381825601754756437575193, −1.96424090556558547088039419660, −1.0026960115701822027813320121,
1.09220690101753291397213359011, 2.37793171019495595665718229505, 3.41969946168640093654227482833, 4.02012195061752569636833810468, 5.80963033097888207021578578960, 6.77075939627869593142876080995, 7.61939347735244688077309946773, 8.63289430280723027950439291928, 9.41613110083320439434795432661, 10.581814677114980126827287182, 11.32395407332523395216798186112, 12.54058369747664717504582245338, 13.57691601735636690477171503248, 14.35707592960489839270715673808, 14.83158985258994317364354840486, 15.93725992155015402746657410961, 16.7990478294082717244127401609, 18.22643927231220727601499076262, 18.831844515698597920827098116506, 19.30326558145927415029881887881, 20.46706864905794537324113058612, 21.22763516029022484754944254580, 22.00783393894118025186200581955, 23.084011245334319986181267480180, 23.80543072844659081206017188972