| L(s) = 1 | − 1.73·3-s − 7-s + 1.99·9-s + 1.73·19-s + 1.73·21-s + 23-s + 25-s − 1.73·27-s − 29-s + 1.73·41-s + 1.73·47-s + 49-s − 53-s − 2.99·57-s − 1.99·63-s + 67-s − 1.73·69-s − 71-s − 1.73·73-s − 1.73·75-s + 0.999·81-s + 1.73·87-s − 1.73·89-s + 1.73·97-s − 1.73·101-s − 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 7-s + 1.99·9-s + 1.73·19-s + 1.73·21-s + 23-s + 25-s − 1.73·27-s − 29-s + 1.73·41-s + 1.73·47-s + 49-s − 53-s − 2.99·57-s − 1.99·63-s + 67-s − 1.73·69-s − 71-s − 1.73·73-s − 1.73·75-s + 0.999·81-s + 1.73·87-s − 1.73·89-s + 1.73·97-s − 1.73·101-s − 107-s + 109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5205691351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5205691351\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 1.73T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - 1.73T + T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.73T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 1.73T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + 1.73T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.73T + T^{2} \) |
| 97 | \( 1 - 1.73T + T^{2} \) |
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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
−10.64412267982860464758153320361, −9.733817669010682121482022605962, −9.096389976472646023040836909190, −7.42706365991508106006628374285, −6.93561321851813590221331096498, −5.91043855455644174829468058397, −5.39650767086687713977134317255, −4.32668210146249067638925109065, −3.04287168102832605721011115747, −0.982292206454670002103035977039,
0.982292206454670002103035977039, 3.04287168102832605721011115747, 4.32668210146249067638925109065, 5.39650767086687713977134317255, 5.91043855455644174829468058397, 6.93561321851813590221331096498, 7.42706365991508106006628374285, 9.096389976472646023040836909190, 9.733817669010682121482022605962, 10.64412267982860464758153320361
