| L(s) = 1 | + 1.73·3-s + 4-s − 5-s + 1.99·9-s − 11-s + 1.73·12-s − 1.73·13-s − 1.73·15-s + 16-s − 19-s − 20-s − 1.73·23-s + 25-s + 1.73·27-s + 29-s + 31-s − 1.73·33-s + 1.99·36-s − 2.99·39-s − 44-s − 1.99·45-s + 1.73·48-s − 49-s − 1.73·52-s + 55-s − 1.73·57-s + 59-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 4-s − 5-s + 1.99·9-s − 11-s + 1.73·12-s − 1.73·13-s − 1.73·15-s + 16-s − 19-s − 20-s − 1.73·23-s + 25-s + 1.73·27-s + 29-s + 31-s − 1.73·33-s + 1.99·36-s − 2.99·39-s − 44-s − 1.99·45-s + 1.73·48-s − 49-s − 1.73·52-s + 55-s − 1.73·57-s + 59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.579539790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.579539790\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 151 | \( 1 + T \) |
| good | 2 | \( 1 - T^{2} \) |
| 3 | \( 1 - 1.73T + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + 1.73T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + 1.73T + T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.73T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.73T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.73T + T^{2} \) |
| 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
−10.21782193389046636105925250954, −9.896451532295190443474116702117, −8.434366475592738567846902781015, −8.019490309682989803793419777293, −7.42377890726675313317546394023, −6.56815152140035116736119583521, −4.86767336694668303350570503257, −3.84126309679339668186302161872, −2.73707644383664244560233770965, −2.23105790776941460457164632467,
2.23105790776941460457164632467, 2.73707644383664244560233770965, 3.84126309679339668186302161872, 4.86767336694668303350570503257, 6.56815152140035116736119583521, 7.42377890726675313317546394023, 8.019490309682989803793419777293, 8.434366475592738567846902781015, 9.896451532295190443474116702117, 10.21782193389046636105925250954
