L(s) = 1 | + (−0.5 + 0.866i)3-s + (2.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (2 − 3.46i)11-s − 7·13-s + (−3 + 5.19i)17-s + (−1.5 − 2.59i)19-s + (−2 + 1.73i)21-s + (−1 − 1.73i)23-s + 0.999·27-s − 2·29-s + (−3.5 + 6.06i)31-s + (1.99 + 3.46i)33-s + (−3.5 − 6.06i)37-s + (3.5 − 6.06i)39-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (0.944 + 0.327i)7-s + (−0.166 − 0.288i)9-s + (0.603 − 1.04i)11-s − 1.94·13-s + (−0.727 + 1.26i)17-s + (−0.344 − 0.596i)19-s + (−0.436 + 0.377i)21-s + (−0.208 − 0.361i)23-s + 0.192·27-s − 0.371·29-s + (−0.628 + 1.08i)31-s + (0.348 + 0.603i)33-s + (−0.575 − 0.996i)37-s + (0.560 − 0.970i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2273620935\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2273620935\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
good | 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 7T + 13T^{2} \) |
| 17 | \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 + 2.59i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 + (-5 - 8.66i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4 - 6.92i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-7.5 + 12.9i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + (1 + 1.73i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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
−9.290377417506438984069517528207, −8.874577922899025839225112512054, −8.072391018049702835978302864352, −7.15873230000105207089665916222, −6.28840306202623242479922454902, −5.41221993190840802185485643486, −4.72884008623077598003933980358, −3.94993468587739925831538477320, −2.75219135134815929153780021482, −1.67590353849184536299715756492,
0.07679757999295594396782435624, 1.73975773157167320899736480188, 2.35243948666619720625065800980, 3.88886856892714354305098245195, 4.92860053430613601880290636904, 5.16939191081595678226258513004, 6.64736568255872030241351507609, 7.15026544126458787510952111352, 7.72011949145626208806275732811, 8.590136166383042391816777086716