| L(s) = 1 | − 2-s + 4-s + 5-s − 1.73·7-s − 8-s − 10-s + 1.73·13-s + 1.73·14-s + 16-s + 19-s + 20-s − 23-s + 25-s − 1.73·26-s − 1.73·28-s − 32-s − 1.73·35-s − 38-s − 40-s − 1.73·41-s + 46-s + 47-s + 1.99·49-s − 50-s + 1.73·52-s + 53-s + 1.73·56-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 5-s − 1.73·7-s − 8-s − 10-s + 1.73·13-s + 1.73·14-s + 16-s + 19-s + 20-s − 23-s + 25-s − 1.73·26-s − 1.73·28-s − 32-s − 1.73·35-s − 38-s − 40-s − 1.73·41-s + 46-s + 47-s + 1.99·49-s − 50-s + 1.73·52-s + 53-s + 1.73·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8682824770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8682824770\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 1.73T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - 1.73T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + 1.73T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 - 1.73T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 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
−8.897890477540212442963696653360, −8.397375180912137666304748937216, −7.21853567200084208870066654996, −6.61138011113362149162249830833, −6.01154955992664242009573294743, −5.52496360970558548656805260817, −3.74303788260714063539041396806, −3.15290028292440029218077555315, −2.13504834424517437338992909118, −0.978875688099410759521335671866,
0.978875688099410759521335671866, 2.13504834424517437338992909118, 3.15290028292440029218077555315, 3.74303788260714063539041396806, 5.52496360970558548656805260817, 6.01154955992664242009573294743, 6.61138011113362149162249830833, 7.21853567200084208870066654996, 8.397375180912137666304748937216, 8.897890477540212442963696653360
