L(s) = 1 | + (1.28 + 0.583i)2-s − 1.66i·3-s + (1.31 + 1.50i)4-s + (0.969 − 2.13i)6-s + 2.84i·7-s + (0.821 + 2.70i)8-s + 0.242·9-s + 5.38i·11-s + (2.49 − 2.18i)12-s + (−1.66 + 3.66i)14-s + (−0.521 + 3.96i)16-s + (0.312 + 0.141i)18-s − 5.77i·19-s + 4.72·21-s + (−3.14 + 6.94i)22-s + ⋯ |
L(s) = 1 | + (0.910 + 0.412i)2-s − 0.958i·3-s + (0.659 + 0.751i)4-s + (0.395 − 0.873i)6-s + 1.07i·7-s + (0.290 + 0.956i)8-s + 0.0808·9-s + 1.62i·11-s + (0.720 − 0.632i)12-s + (−0.444 + 0.980i)14-s + (−0.130 + 0.991i)16-s + (0.0736 + 0.0333i)18-s − 1.32i·19-s + 1.03·21-s + (−0.670 + 1.47i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.48345 + 0.934816i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.48345 + 0.934816i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.28 - 0.583i)T \) |
| 167 | \( 1 - 12.9iT \) |
good | 3 | \( 1 + 1.66iT - 3T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 2.84iT - 7T^{2} \) |
| 11 | \( 1 - 5.38iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 5.77iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 1.43T + 29T^{2} \) |
| 31 | \( 1 + 5.69iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 4.31iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−10.91890801471649885027465779597, −9.622250239961846772076009171490, −8.668159335937056252174674277906, −7.59362801095567040114681852001, −7.02283131688186023953195317919, −6.26062806768974372662438928838, −5.15672030388412105179565918614, −4.35197664073350861929312592585, −2.71045208601058045875129151499, −1.92609200096923150130703102698,
1.20866350310800929421114606735, 3.19055914892390262912693447112, 3.78402410710002792109086936784, 4.68654580494891356293122697500, 5.65521517643522385497610864171, 6.60962227878740515018728869265, 7.68525211243496167721405730512, 8.905142794253590281966904779786, 9.934515122710582246777735201797, 10.71104912941005139571455126432