L(s) = 1 | − i·3-s − i·5-s + 4·7-s − 9-s − 6i·11-s − 4i·13-s − 15-s + 4·17-s − 8i·19-s − 4i·21-s − 25-s + i·27-s + 2i·29-s − 2·31-s − 6·33-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.447i·5-s + 1.51·7-s − 0.333·9-s − 1.80i·11-s − 1.10i·13-s − 0.258·15-s + 0.970·17-s − 1.83i·19-s − 0.872i·21-s − 0.200·25-s + 0.192i·27-s + 0.371i·29-s − 0.359·31-s − 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.187083401\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.187083401\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + iT \) |
good | 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 + 6iT - 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 8iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 14iT - 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 + 6iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−8.039666134224819326657574850248, −7.83404148047962015305520992154, −6.79719896884737015647471624906, −5.75989367912481477456645601131, −5.35306737577611082324714619967, −4.59711094697763103059528579640, −3.36776588174198981797457881723, −2.63263297297571857261669213118, −1.30675375364919074200395392957, −0.67421180068614610644736379463,
1.69564691513700596082837549604, 2.03942475980003325487436959688, 3.57438440489757377713380961268, 4.21086090861768093846247918880, 4.95998097630410274223289650394, 5.55629340031656967926426925146, 6.67515825770868494912661589108, 7.37751957243373832322384289071, 8.021707402237791446808671598682, 8.659305854886755643157009638577