L(s) = 1 | + (−0.939 + 0.342i)5-s + (−0.5 − 0.866i)7-s + (−0.5 + 0.866i)11-s + (0.766 − 0.642i)13-s + (−0.173 + 0.984i)17-s + (0.939 + 0.342i)23-s + (0.766 − 0.642i)25-s + (−0.173 − 0.984i)29-s + (0.5 + 0.866i)31-s + (0.766 + 0.642i)35-s + 37-s + (0.766 + 0.642i)41-s + (0.939 − 0.342i)43-s + (−0.173 − 0.984i)47-s + (−0.5 + 0.866i)49-s + ⋯ |
L(s) = 1 | + (−0.939 + 0.342i)5-s + (−0.5 − 0.866i)7-s + (−0.5 + 0.866i)11-s + (0.766 − 0.642i)13-s + (−0.173 + 0.984i)17-s + (0.939 + 0.342i)23-s + (0.766 − 0.642i)25-s + (−0.173 − 0.984i)29-s + (0.5 + 0.866i)31-s + (0.766 + 0.642i)35-s + 37-s + (0.766 + 0.642i)41-s + (0.939 − 0.342i)43-s + (−0.173 − 0.984i)47-s + (−0.5 + 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9869048338 + 0.1736858768i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9869048338 + 0.1736858768i\) |
\(L(1)\) |
\(\approx\) |
\(0.8894532276 + 0.04490964256i\) |
\(L(1)\) |
\(\approx\) |
\(0.8894532276 + 0.04490964256i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + (-0.939 + 0.342i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.766 - 0.642i)T \) |
| 17 | \( 1 + (-0.173 + 0.984i)T \) |
| 23 | \( 1 + (0.939 + 0.342i)T \) |
| 29 | \( 1 + (-0.173 - 0.984i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.766 + 0.642i)T \) |
| 43 | \( 1 + (0.939 - 0.342i)T \) |
| 47 | \( 1 + (-0.173 - 0.984i)T \) |
| 53 | \( 1 + (0.939 + 0.342i)T \) |
| 59 | \( 1 + (-0.173 + 0.984i)T \) |
| 61 | \( 1 + (0.939 + 0.342i)T \) |
| 67 | \( 1 + (0.173 + 0.984i)T \) |
| 71 | \( 1 + (-0.939 + 0.342i)T \) |
| 73 | \( 1 + (0.766 + 0.642i)T \) |
| 79 | \( 1 + (-0.766 - 0.642i)T \) |
| 83 | \( 1 + (-0.5 - 0.866i)T \) |
| 89 | \( 1 + (0.766 - 0.642i)T \) |
| 97 | \( 1 + (-0.173 + 0.984i)T \) |
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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
−23.96489288720243146125399074111, −23.03754030406345633741563296019, −22.32716540526512415472483687344, −21.2349315944781731999630266418, −20.593112549554597614868178378229, −19.43301993724297076349066123627, −18.8309156245586942554684049331, −18.189444357959010011949761769013, −16.67244053077324853484777050827, −16.06461562089397903198140739850, −15.49316946515393877110744640561, −14.38404385899899743635445438388, −13.25858156266304898533637371479, −12.52540053139885149551280648642, −11.470871040220795323698530312184, −10.97678814507729318053223621309, −9.37943056675173664842835578418, −8.783680481943224370565740028602, −7.852175228953988693642812498760, −6.72069445612639427594358301177, −5.683759053700695012664101364598, −4.64385063295959605144740921142, −3.4759785441446192845452662607, −2.56456850663877354835877120565, −0.77621634538398889377311684717,
0.96679794718161018833885638944, 2.69972073670797114248704880198, 3.74920157489290622691553140855, 4.49069095327783854759611874482, 5.9362408725906715609566334252, 7.03508622275038217042206279958, 7.70355324319545260328973678596, 8.67518601539143562839749807809, 10.06971329984141249509106280837, 10.65734885830681509934179611001, 11.59074334007802746931528849598, 12.783184404174393772585481740, 13.28985385619123885943769840626, 14.61942863010990962283135068730, 15.381671647156711065761372559971, 16.06445677474076825247468202031, 17.13382720733105853531117386209, 17.97003720804796923557588960318, 19.01228701101963862020058318840, 19.73955285283073448330863948021, 20.416528875078116180070413767218, 21.37026662299653420281782304812, 22.67290174416075630755238493647, 23.14427333644157274098716571548, 23.64912085901855834452299537768