L(s) = 1 | + 2-s + 4-s + (−0.648 + 2.56i)7-s + 8-s + 1.29i·11-s + 3.13·13-s + (−0.648 + 2.56i)14-s + 16-s − 5.53i·17-s + 7.37i·19-s + 1.29i·22-s + 1.83·23-s + 3.13·26-s + (−0.648 + 2.56i)28-s + 1.83i·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + (−0.245 + 0.969i)7-s + 0.353·8-s + 0.390i·11-s + 0.868·13-s + (−0.173 + 0.685i)14-s + 0.250·16-s − 1.34i·17-s + 1.69i·19-s + 0.276i·22-s + 0.382·23-s + 0.613·26-s + (−0.122 + 0.484i)28-s + 0.340i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.824174831\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.824174831\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.648 - 2.56i)T \) |
good | 11 | \( 1 - 1.29iT - 11T^{2} \) |
| 13 | \( 1 - 3.13T + 13T^{2} \) |
| 17 | \( 1 + 5.53iT - 17T^{2} \) |
| 19 | \( 1 - 7.37iT - 19T^{2} \) |
| 23 | \( 1 - 1.83T + 23T^{2} \) |
| 29 | \( 1 - 1.83iT - 29T^{2} \) |
| 31 | \( 1 + 10.4iT - 31T^{2} \) |
| 37 | \( 1 - 10.6iT - 37T^{2} \) |
| 41 | \( 1 - 3.13T + 41T^{2} \) |
| 43 | \( 1 - 3.53iT - 43T^{2} \) |
| 47 | \( 1 - 10.7iT - 47T^{2} \) |
| 53 | \( 1 - 4.42T + 53T^{2} \) |
| 59 | \( 1 + 7.18T + 59T^{2} \) |
| 61 | \( 1 - 4.88iT - 61T^{2} \) |
| 67 | \( 1 - 9.79iT - 67T^{2} \) |
| 71 | \( 1 - 7.37iT - 71T^{2} \) |
| 73 | \( 1 - 3.40T + 73T^{2} \) |
| 79 | \( 1 + 9.01T + 79T^{2} \) |
| 83 | \( 1 + 6.26iT - 83T^{2} \) |
| 89 | \( 1 - 7.94T + 89T^{2} \) |
| 97 | \( 1 - 8.09T + 97T^{2} \) |
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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
−8.803814635193679975769863379303, −7.983894302469709570475743733180, −7.29539162285554192778490316701, −6.21516427019766326417035403054, −5.90993980951876131650415198466, −4.99261931429968298266756195260, −4.18145595610872771541247090974, −3.20966700973152877416588310420, −2.48258886580716257080643522384, −1.33592873431003705149913965815,
0.71474648593895066474248615188, 1.92877478671784693149104763058, 3.20929636462895420238120967839, 3.76547584743762824973487266949, 4.59007296301177499346066648539, 5.45134769323418548018097550705, 6.33975668513855582889052254672, 6.87568084043909075806164770748, 7.61689932676213121346004850160, 8.578542709289957166853880150725