L(s) = 1 | − 2.49i·2-s + 3-s − 4.20·4-s − 1.87i·5-s − 2.49i·6-s − 2.20i·7-s + 5.49i·8-s + 9-s − 4.67·10-s − 3.27·11-s − 4.20·12-s + 4.27·13-s − 5.49·14-s − 1.87i·15-s + 5.27·16-s − 6.36·17-s + ⋯ |
L(s) = 1 | − 1.76i·2-s + 0.577·3-s − 2.10·4-s − 0.839i·5-s − 1.01i·6-s − 0.833i·7-s + 1.94i·8-s + 0.333·9-s − 1.47·10-s − 0.987·11-s − 1.21·12-s + 1.18·13-s − 1.46·14-s − 0.484i·15-s + 1.31·16-s − 1.54·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.363211 + 1.26607i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.363211 + 1.26607i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 157 | \( 1 + (10.6 + 6.64i)T \) |
good | 2 | \( 1 + 2.49iT - 2T^{2} \) |
| 5 | \( 1 + 1.87iT - 5T^{2} \) |
| 7 | \( 1 + 2.20iT - 7T^{2} \) |
| 11 | \( 1 + 3.27T + 11T^{2} \) |
| 13 | \( 1 - 4.27T + 13T^{2} \) |
| 17 | \( 1 + 6.36T + 17T^{2} \) |
| 19 | \( 1 - 2.88T + 19T^{2} \) |
| 23 | \( 1 + 4.80iT - 23T^{2} \) |
| 29 | \( 1 - 5.55iT - 29T^{2} \) |
| 31 | \( 1 + 0.391T + 31T^{2} \) |
| 37 | \( 1 + 5.88T + 37T^{2} \) |
| 41 | \( 1 - 2.68iT - 41T^{2} \) |
| 43 | \( 1 + 4.21iT - 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 8.29iT - 53T^{2} \) |
| 59 | \( 1 + 6.81iT - 59T^{2} \) |
| 61 | \( 1 + 0.933iT - 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 6.35T + 71T^{2} \) |
| 73 | \( 1 + 10.0iT - 73T^{2} \) |
| 79 | \( 1 - 10.5iT - 79T^{2} \) |
| 83 | \( 1 + 0.160iT - 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 + 0.864iT - 97T^{2} \) |
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
−10.74341284619665316112884996169, −9.797602601512020348525198425515, −8.757414347548748514215901035668, −8.444370935316934597789938416971, −6.95262786321517192412083542915, −5.13301821845556993136537893781, −4.27684404196999860250630842133, −3.35738548748156181526824934010, −2.10040889737571115173692170873, −0.78461848929505991659728557632,
2.59236661805174398339778001605, 3.98761379656555240109113721756, 5.30607425850234228630653508653, 6.10535774452777893171370118558, 7.00830834667296233246895079684, 7.78343886013978865322335096129, 8.677911587976483708646949303993, 9.212447441695564438759535273500, 10.44443478857003123730573089952, 11.44260601845055293905458425772