L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.707 − 0.707i)8-s + (0.707 + 1.22i)11-s + (0.500 − 0.866i)16-s + (−0.999 + i)22-s + (−0.707 + 1.22i)23-s + (0.5 + 0.866i)25-s + 1.41i·29-s + (0.965 + 0.258i)32-s − 2i·43-s + (−1.22 − 0.707i)44-s + (−1.36 − 0.366i)46-s + (−0.707 + 0.707i)50-s + (−1.22 + 0.707i)53-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.707 − 0.707i)8-s + (0.707 + 1.22i)11-s + (0.500 − 0.866i)16-s + (−0.999 + i)22-s + (−0.707 + 1.22i)23-s + (0.5 + 0.866i)25-s + 1.41i·29-s + (0.965 + 0.258i)32-s − 2i·43-s + (−1.22 − 0.707i)44-s + (−1.36 − 0.366i)46-s + (−0.707 + 0.707i)50-s + (−1.22 + 0.707i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.114972233\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.114972233\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + 2iT - T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - 1.41T + T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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
−9.454776200705221084807764747374, −9.021529598647636537266141796625, −8.026730228272483717606957010608, −7.19276721689179413991230496520, −6.80681933209717227295141138249, −5.70825225701598322385765906763, −5.01289350689286962193115530776, −4.10092296601415750846695063242, −3.28702009548182394533229471380, −1.67278834907672215413076138761,
0.834678844511689710610614715460, 2.24050145656720541149907748375, 3.18973274159834233120614987943, 4.10393148289898009377031135101, 4.84819029447022482586852992499, 6.07036944814744043290356472993, 6.40614183515116399892389040275, 8.022940120291773869131828362860, 8.483525132410675848436489804199, 9.377124230451240160338569289432