L(s) = 1 | + i·2-s − 4-s − 0.601·5-s − i·8-s − 0.601i·10-s − 1.20i·11-s + 0.649i·13-s + 16-s − 0.564·17-s − 0.249i·19-s + 0.601·20-s + 1.20·22-s + 3.95i·23-s − 4.63·25-s − 0.649·26-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.269·5-s − 0.353i·8-s − 0.190i·10-s − 0.361i·11-s + 0.180i·13-s + 0.250·16-s − 0.136·17-s − 0.0571i·19-s + 0.134·20-s + 0.255·22-s + 0.824i·23-s − 0.927·25-s − 0.127·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7832900346\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7832900346\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 0.601T + 5T^{2} \) |
| 11 | \( 1 + 1.20iT - 11T^{2} \) |
| 13 | \( 1 - 0.649iT - 13T^{2} \) |
| 17 | \( 1 + 0.564T + 17T^{2} \) |
| 19 | \( 1 + 0.249iT - 19T^{2} \) |
| 23 | \( 1 - 3.95iT - 23T^{2} \) |
| 29 | \( 1 + 1.37iT - 29T^{2} \) |
| 31 | \( 1 + 7.11iT - 31T^{2} \) |
| 37 | \( 1 - 5.17T + 37T^{2} \) |
| 41 | \( 1 + 7.32T + 41T^{2} \) |
| 43 | \( 1 - 9.36T + 43T^{2} \) |
| 47 | \( 1 + 9.12T + 47T^{2} \) |
| 53 | \( 1 + 10.3iT - 53T^{2} \) |
| 59 | \( 1 + 7.26T + 59T^{2} \) |
| 61 | \( 1 - 0.865iT - 61T^{2} \) |
| 67 | \( 1 + 9.37T + 67T^{2} \) |
| 71 | \( 1 + 4.28iT - 71T^{2} \) |
| 73 | \( 1 + 15.7iT - 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 5.58T + 83T^{2} \) |
| 89 | \( 1 + 7.62T + 89T^{2} \) |
| 97 | \( 1 + 13.2iT - 97T^{2} \) |
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
−8.566341671026086806399104651444, −7.85152528615338282115009856213, −7.34279857037183448923596260361, −6.31619993434465536940882220373, −5.81592262420081146688969146902, −4.83298441433102329558439903154, −4.04022064270080203058972317268, −3.17252457334199468118026339017, −1.82521182315970535529942616180, −0.27309741594948712048088374875,
1.21617275486729990479977364230, 2.35148814812828278959327425184, 3.25144734894054052088377299152, 4.18700955232236654340513364636, 4.86959682045551665131665707811, 5.83056950186492650589873476505, 6.73271415769998659303963512821, 7.62219021824733671879056245365, 8.330580893215037806480885345569, 9.077619051055099370885674326268