L(s) = 1 | + 1.73i·5-s + 4.58i·11-s + 3.46i·13-s + 5.19i·17-s + 2.64·19-s − 4.58i·23-s + 2.00·25-s − 2.64·31-s + 7·37-s + 3.46i·41-s + 9.16i·43-s + 7.93·47-s − 3·53-s − 7.93·55-s + 7.93·59-s + ⋯ |
L(s) = 1 | + 0.774i·5-s + 1.38i·11-s + 0.960i·13-s + 1.26i·17-s + 0.606·19-s − 0.955i·23-s + 0.400·25-s − 0.475·31-s + 1.15·37-s + 0.541i·41-s + 1.39i·43-s + 1.15·47-s − 0.412·53-s − 1.07·55-s + 1.03·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.780970414\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.780970414\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 11 | \( 1 - 4.58iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 5.19iT - 17T^{2} \) |
| 19 | \( 1 - 2.64T + 19T^{2} \) |
| 23 | \( 1 + 4.58iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 2.64T + 31T^{2} \) |
| 37 | \( 1 - 7T + 37T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 9.16iT - 43T^{2} \) |
| 47 | \( 1 - 7.93T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 - 7.93T + 59T^{2} \) |
| 61 | \( 1 - 1.73iT - 61T^{2} \) |
| 67 | \( 1 - 4.58iT - 67T^{2} \) |
| 71 | \( 1 + 9.16iT - 71T^{2} \) |
| 73 | \( 1 - 5.19iT - 73T^{2} \) |
| 79 | \( 1 - 4.58iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 1.73iT - 89T^{2} \) |
| 97 | \( 1 + 3.46iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.068197265573359416087335590756, −7.46081516011877595560406735258, −6.72314305445480250859166033094, −6.37937103308212979543152835097, −5.41070402845889236620506200437, −4.43903261586478460751474890210, −4.06294150397276758023733919997, −2.93525286129226448424257684494, −2.22266205350718119884847835774, −1.32368125654640021219950217311,
0.49287259945042488349503557763, 1.10769675307881385490225290580, 2.50241765438592338457898545696, 3.26173176890747296441138653102, 3.96819414719334244870961484934, 5.14210947896741234214840893680, 5.37328300468896328145378727562, 6.09987262996344609563937585969, 7.12856286254890046890613340841, 7.70116082095266431051247133867