L(s) = 1 | + (1.22 − 0.707i)2-s + (−0.866 − 1.5i)3-s + (0.999 − 1.73i)4-s + (0.448 − 2.19i)5-s + (−2.12 − 1.22i)6-s + (1.22 + 2.12i)7-s − 2.82i·8-s + (−1.5 + 2.59i)9-s + (−1 − 2.99i)10-s + (4.5 − 2.59i)11-s − 3.46·12-s + (−1.22 + 2.12i)13-s + (3 + 1.73i)14-s + (−3.67 + 1.22i)15-s + (−2.00 − 3.46i)16-s − 5.19·17-s + ⋯ |
L(s) = 1 | + (0.866 − 0.499i)2-s + (−0.499 − 0.866i)3-s + (0.499 − 0.866i)4-s + (0.200 − 0.979i)5-s + (−0.866 − 0.499i)6-s + (0.462 + 0.801i)7-s − 0.999i·8-s + (−0.5 + 0.866i)9-s + (−0.316 − 0.948i)10-s + (1.35 − 0.783i)11-s − 0.999·12-s + (−0.339 + 0.588i)13-s + (0.801 + 0.462i)14-s + (−0.948 + 0.316i)15-s + (−0.500 − 0.866i)16-s − 1.26·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.523 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.523 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.963658 - 1.72302i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.963658 - 1.72302i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.22 + 0.707i)T \) |
| 3 | \( 1 + (0.866 + 1.5i)T \) |
| 5 | \( 1 + (-0.448 + 2.19i)T \) |
good | 7 | \( 1 + (-1.22 - 2.12i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.5 + 2.59i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.22 - 2.12i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 5.19T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + (-2.44 - 1.41i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.24 - 7.34i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-6.36 - 3.67i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + (-1.5 - 0.866i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-7.79 + 4.5i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.12 + 3.53i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 1.41iT - 53T^{2} \) |
| 59 | \( 1 + (1.5 + 0.866i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.36 - 3.67i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.59 + 1.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4.24T + 71T^{2} \) |
| 73 | \( 1 - 3iT - 73T^{2} \) |
| 79 | \( 1 + (6.36 - 3.67i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 10.3iT - 89T^{2} \) |
| 97 | \( 1 + (2.59 - 1.5i)T + (48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.50357419031564585617810037411, −10.68109568476491866102788597686, −9.049394770646899439140310470881, −8.614422914668273208760481664838, −6.85833003542518615374777737690, −6.17359629784165468941603973386, −5.17216754070877642297483405846, −4.28250862111304564469436614638, −2.36283507648120124814885832397, −1.25324714465702152414865622242,
2.61176215019970971255312477391, 4.21627611287083045576928552820, 4.43051821861483150243112615546, 6.12659101849243361387963669145, 6.62149440985234224811382218036, 7.67538787914853956745910597293, 9.029959027864327537257085335768, 10.20093789890704011155460987191, 10.96523295028966561185202389687, 11.61866547984322004287301354772