| L(s) = 1 | − 1.73·3-s + 5-s + 1.73·7-s + 1.99·9-s + 13-s − 1.73·15-s − 17-s − 2.99·21-s − 1.73·27-s + 1.73·35-s − 37-s − 1.73·39-s + 1.73·43-s + 1.99·45-s − 1.73·47-s + 1.99·49-s + 1.73·51-s + 3.46·63-s + 65-s + 1.73·71-s + 0.999·81-s − 85-s + 1.73·91-s − 2.99·105-s − 109-s + 1.73·111-s − 2·113-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 5-s + 1.73·7-s + 1.99·9-s + 13-s − 1.73·15-s − 17-s − 2.99·21-s − 1.73·27-s + 1.73·35-s − 37-s − 1.73·39-s + 1.73·43-s + 1.99·45-s − 1.73·47-s + 1.99·49-s + 1.73·51-s + 3.46·63-s + 65-s + 1.73·71-s + 0.999·81-s − 85-s + 1.73·91-s − 2.99·105-s − 109-s + 1.73·111-s − 2·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9602224968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9602224968\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 1.73T + T^{2} \) |
| 5 | \( 1 - T + T^{2} \) |
| 7 | \( 1 - 1.73T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.73T + T^{2} \) |
| 47 | \( 1 + 1.73T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 1.73T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 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
−9.743253609200885047431461506841, −8.793448751312110194453327117110, −7.920898222365619710477732598673, −6.86764377975527281420949359514, −6.17314775983865868081456904895, −5.46460819965475710428034955340, −4.90019821984642334388702798911, −4.07367897868262692162655780286, −2.05562324370534467128878200204, −1.25639394657444950735104589609,
1.25639394657444950735104589609, 2.05562324370534467128878200204, 4.07367897868262692162655780286, 4.90019821984642334388702798911, 5.46460819965475710428034955340, 6.17314775983865868081456904895, 6.86764377975527281420949359514, 7.920898222365619710477732598673, 8.793448751312110194453327117110, 9.743253609200885047431461506841
