L(s) = 1 | + (1 + i)2-s − 2i·3-s + 2i·4-s + (1 − 2i)5-s + (2 − 2i)6-s + (−3 + 3i)7-s + (−2 + 2i)8-s − 9-s + (3 − i)10-s + (−1 + i)11-s + 4·12-s − 2·13-s − 6·14-s + (−4 − 2i)15-s − 4·16-s + (1 − i)17-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s − 1.15i·3-s + i·4-s + (0.447 − 0.894i)5-s + (0.816 − 0.816i)6-s + (−1.13 + 1.13i)7-s + (−0.707 + 0.707i)8-s − 0.333·9-s + (0.948 − 0.316i)10-s + (−0.301 + 0.301i)11-s + 1.15·12-s − 0.554·13-s − 1.60·14-s + (−1.03 − 0.516i)15-s − 16-s + (0.242 − 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.25124 + 0.100864i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.25124 + 0.100864i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - i)T \) |
| 5 | \( 1 + (-1 + 2i)T \) |
good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 + (3 - 3i)T - 7iT^{2} \) |
| 11 | \( 1 + (1 - i)T - 11iT^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (-1 + i)T - 17iT^{2} \) |
| 19 | \( 1 + (-3 + 3i)T - 19iT^{2} \) |
| 23 | \( 1 + (1 + i)T + 23iT^{2} \) |
| 29 | \( 1 + (-7 - 7i)T + 29iT^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 4iT - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (-7 - 7i)T + 47iT^{2} \) |
| 53 | \( 1 + 8iT - 53T^{2} \) |
| 59 | \( 1 + (3 + 3i)T + 59iT^{2} \) |
| 61 | \( 1 + (1 - i)T - 61iT^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (3 - 3i)T - 73iT^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + (11 - 11i)T - 97iT^{2} \) |
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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
−14.15112524404569176674908270993, −13.18539645043385362545486359499, −12.49276997095376938063047430271, −12.06861790428375561600779554210, −9.592938614681951837097941131581, −8.524464693192186877471115792571, −7.19877950386906355931543697614, −6.16330575759644388198874314227, −5.04034024022624568110967431733, −2.64535399108189045264510170835,
3.10705146482163524261049807370, 4.09778591592978932596819751838, 5.73022578469574864023764171080, 7.06035091297824136120629226442, 9.563440182256676483416231456398, 10.22068358543928838428592717954, 10.66720740706958347951025042831, 12.18801456944399837843399013207, 13.56263174322409099297897186637, 14.13143550792508788546022118053