L(s) = 1 | + 1.73·5-s + 7-s − 1.73·11-s − 19-s + 1.73·23-s + 1.99·25-s + 31-s + 1.73·35-s + 37-s − 1.73·41-s + 49-s − 2.99·55-s − 1.73·71-s − 1.73·77-s − 1.73·89-s − 1.73·95-s + 103-s − 109-s + 2.99·115-s + ⋯ |
L(s) = 1 | + 1.73·5-s + 7-s − 1.73·11-s − 19-s + 1.73·23-s + 1.99·25-s + 31-s + 1.73·35-s + 37-s − 1.73·41-s + 49-s − 2.99·55-s − 1.73·71-s − 1.73·77-s − 1.73·89-s − 1.73·95-s + 103-s − 109-s + 2.99·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.762371842\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.762371842\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 1.73T + T^{2} \) |
| 11 | \( 1 + 1.73T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 - 1.73T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 + 1.73T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 1.73T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.73T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.803399672344762779713323452016, −8.323523457831417576498784910239, −7.40183849215642323679088257812, −6.55683909552792225655884641118, −5.72523199243151514712253076542, −5.10971966272450512138297495968, −4.59853894110085627561041649773, −2.90380648745626631260735297120, −2.33167326097782463055079011489, −1.36546107278891605610404738331,
1.36546107278891605610404738331, 2.33167326097782463055079011489, 2.90380648745626631260735297120, 4.59853894110085627561041649773, 5.10971966272450512138297495968, 5.72523199243151514712253076542, 6.55683909552792225655884641118, 7.40183849215642323679088257812, 8.323523457831417576498784910239, 8.803399672344762779713323452016