L(s) = 1 | + 1.77i·5-s + 0.423·7-s − 1.66i·11-s + 6.39i·13-s + 17-s + 7.19i·19-s + 5.68·23-s + 1.84·25-s + 3.37i·29-s − 0.266·31-s + 0.751i·35-s − 3.50i·37-s − 8.79·41-s − 7.53i·43-s + 2.10·47-s + ⋯ |
L(s) = 1 | + 0.794i·5-s + 0.159·7-s − 0.501i·11-s + 1.77i·13-s + 0.242·17-s + 1.65i·19-s + 1.18·23-s + 0.369·25-s + 0.625i·29-s − 0.0478·31-s + 0.127i·35-s − 0.576i·37-s − 1.37·41-s − 1.14i·43-s + 0.307·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.576 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.576 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.613057608\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.613057608\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - 1.77iT - 5T^{2} \) |
| 7 | \( 1 - 0.423T + 7T^{2} \) |
| 11 | \( 1 + 1.66iT - 11T^{2} \) |
| 13 | \( 1 - 6.39iT - 13T^{2} \) |
| 19 | \( 1 - 7.19iT - 19T^{2} \) |
| 23 | \( 1 - 5.68T + 23T^{2} \) |
| 29 | \( 1 - 3.37iT - 29T^{2} \) |
| 31 | \( 1 + 0.266T + 31T^{2} \) |
| 37 | \( 1 + 3.50iT - 37T^{2} \) |
| 41 | \( 1 + 8.79T + 41T^{2} \) |
| 43 | \( 1 + 7.53iT - 43T^{2} \) |
| 47 | \( 1 - 2.10T + 47T^{2} \) |
| 53 | \( 1 + 3.32iT - 53T^{2} \) |
| 59 | \( 1 + 3.98iT - 59T^{2} \) |
| 61 | \( 1 - 9.74iT - 61T^{2} \) |
| 67 | \( 1 + 4.82iT - 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 - 9.68T + 79T^{2} \) |
| 83 | \( 1 - 8.81iT - 83T^{2} \) |
| 89 | \( 1 + 6.39T + 89T^{2} \) |
| 97 | \( 1 - 3.41T + 97T^{2} \) |
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
−8.563993330170162240619863368328, −7.73899714563579199116144847584, −6.87780061951413642428057081629, −6.59679135439896539522273851879, −5.63130966245191189895736803918, −4.86838645588633866402963980728, −3.84408996339171886700115475784, −3.31622692192756472959747551118, −2.21023816979966593871856720169, −1.34580757025482094108847821405,
0.46120380812367962332998061792, 1.34783477818710872288189715789, 2.69200124158310156087100030232, 3.27877413641006537125806523224, 4.62877281902401537105331284417, 4.92824023296297709572394040454, 5.66113483139899553160952668107, 6.64793859270966459287927018050, 7.33930832530730346182440343089, 8.128944110942119130609650553355