| L(s) = 1 | + 2-s + 1.73·3-s + 1.73·6-s − 8-s + 1.99·9-s − 1.73·13-s − 16-s + 1.99·18-s + 23-s − 1.73·24-s + 25-s − 1.73·26-s + 1.73·27-s − 29-s − 1.73·31-s − 2.99·39-s − 1.73·41-s + 46-s + 1.73·47-s − 1.73·48-s + 50-s + 1.73·54-s − 58-s − 1.73·62-s + 64-s + 1.73·69-s − 71-s + ⋯ |
| L(s) = 1 | + 2-s + 1.73·3-s + 1.73·6-s − 8-s + 1.99·9-s − 1.73·13-s − 16-s + 1.99·18-s + 23-s − 1.73·24-s + 25-s − 1.73·26-s + 1.73·27-s − 29-s − 1.73·31-s − 2.99·39-s − 1.73·41-s + 46-s + 1.73·47-s − 1.73·48-s + 50-s + 1.73·54-s − 58-s − 1.73·62-s + 64-s + 1.73·69-s − 71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.400569811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.400569811\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - T + T^{2} \) |
| 3 | \( 1 - 1.73T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.73T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + 1.73T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.73T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 1.73T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 - 1.73T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−9.669163541510414716133438503487, −9.179965275849793451652471011900, −8.536098643059979346920811162565, −7.43486447586164643800255959229, −6.95365250957126660093384655387, −5.41583107225550668934781340970, −4.69015119018573074482191805952, −3.71649043049436442047744266606, −2.98411339137512860393557114936, −2.12057645981206078869168367361,
2.12057645981206078869168367361, 2.98411339137512860393557114936, 3.71649043049436442047744266606, 4.69015119018573074482191805952, 5.41583107225550668934781340970, 6.95365250957126660093384655387, 7.43486447586164643800255959229, 8.536098643059979346920811162565, 9.179965275849793451652471011900, 9.669163541510414716133438503487
