| L(s) = 1 | − 1.73·2-s + 1.99·4-s − 7-s − 1.73·8-s + 1.73·11-s + 1.73·14-s + 0.999·16-s − 2.99·22-s + 25-s − 1.99·28-s + 37-s + 43-s + 3.46·44-s + 49-s − 1.73·50-s + 1.73·53-s + 1.73·56-s − 1.00·64-s − 67-s − 1.73·71-s − 1.73·74-s − 1.73·77-s − 79-s − 1.73·86-s − 2.99·88-s − 1.73·98-s + 1.99·100-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 1.99·4-s − 7-s − 1.73·8-s + 1.73·11-s + 1.73·14-s + 0.999·16-s − 2.99·22-s + 25-s − 1.99·28-s + 37-s + 43-s + 3.46·44-s + 49-s − 1.73·50-s + 1.73·53-s + 1.73·56-s − 1.00·64-s − 67-s − 1.73·71-s − 1.73·74-s − 1.73·77-s − 79-s − 1.73·86-s − 2.99·88-s − 1.73·98-s + 1.99·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4159967336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4159967336\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + 1.73T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.73T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 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 | \( 1 - T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.73T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + 1.73T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 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
−10.65655710474212145386142787143, −9.869837757445742534581493883818, −9.135096920506841633084177805828, −8.712448766472471633899433084785, −7.43575546979320228008489387266, −6.74613410204020087286201809739, −6.01180283237861809057355223694, −4.09182102344938789480164144096, −2.73761425256176880608674184749, −1.17964753789132369652410274054,
1.17964753789132369652410274054, 2.73761425256176880608674184749, 4.09182102344938789480164144096, 6.01180283237861809057355223694, 6.74613410204020087286201809739, 7.43575546979320228008489387266, 8.712448766472471633899433084785, 9.135096920506841633084177805828, 9.869837757445742534581493883818, 10.65655710474212145386142787143
