L(s) = 1 | + 2.56i·5-s − i·7-s + 1.15·11-s + 0.578·13-s − 5.39i·17-s + 6.20i·19-s − 7.62·23-s − 1.57·25-s − 1.41i·29-s + 5.04i·31-s + 2.56·35-s − 9.83·37-s + 6.21i·41-s + 11.2i·43-s + 11.0·47-s + ⋯ |
L(s) = 1 | + 1.14i·5-s − 0.377i·7-s + 0.346·11-s + 0.160·13-s − 1.30i·17-s + 1.42i·19-s − 1.58·23-s − 0.315·25-s − 0.262i·29-s + 0.906i·31-s + 0.433·35-s − 1.61·37-s + 0.970i·41-s + 1.71i·43-s + 1.61·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.015417558\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.015417558\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
good | 5 | \( 1 - 2.56iT - 5T^{2} \) |
| 11 | \( 1 - 1.15T + 11T^{2} \) |
| 13 | \( 1 - 0.578T + 13T^{2} \) |
| 17 | \( 1 + 5.39iT - 17T^{2} \) |
| 19 | \( 1 - 6.20iT - 19T^{2} \) |
| 23 | \( 1 + 7.62T + 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 5.04iT - 31T^{2} \) |
| 37 | \( 1 + 9.83T + 37T^{2} \) |
| 41 | \( 1 - 6.21iT - 41T^{2} \) |
| 43 | \( 1 - 11.2iT - 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 4.53iT - 53T^{2} \) |
| 59 | \( 1 + 4.83T + 59T^{2} \) |
| 61 | \( 1 + 0.951T + 61T^{2} \) |
| 67 | \( 1 + 2.78iT - 67T^{2} \) |
| 71 | \( 1 + 3.68T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 - 12.8iT - 79T^{2} \) |
| 83 | \( 1 + 8.77T + 83T^{2} \) |
| 89 | \( 1 + 5.68iT - 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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
−8.639712757597165935848855956305, −7.87323297130115347487316062509, −7.26004692722898980882350922399, −6.55405632206857847108717100837, −5.97838784709680202788639144234, −5.00151657578354335755051840934, −3.99495935530644060217770613224, −3.35023215003655398456363467806, −2.47977778246253514303772233892, −1.36891454208636501227539696840,
0.28775773210053779626743692482, 1.54286010113212644936170196913, 2.37971237986761298207558292979, 3.75004434274696007627680273143, 4.27131177856948784425301766921, 5.24921193929071043720371649670, 5.78112542337493407347588125024, 6.62849825400165655626852407153, 7.49912354747355208696218323431, 8.370300712589836000329262014342