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 − 1.73·23-s + 1.99·28-s + 1.73·29-s − 37-s − 43-s − 3.46·44-s − 2.99·46-s + 49-s + 1.73·56-s + 2.99·58-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 − 3.46·92-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 − 1.73·23-s + 1.99·28-s + 1.73·29-s − 37-s − 43-s − 3.46·44-s − 2.99·46-s + 49-s + 1.73·56-s + 2.99·58-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 − 3.46·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.919340420\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.919340420\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 2 | \( 1 - 1.73T + T^{2} \) |
| 11 | \( 1 + 1.73T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.73T + T^{2} \) |
| 29 | \( 1 - 1.73T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 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
−10.02944591943552765476781696696, −8.424769593532622569408273807934, −7.932734379283643434734382269335, −6.99940598940526418624129680332, −6.05763279934618195094641238203, −5.23210941933012756997585437601, −4.79922409943941392386911840550, −3.84928487291438368783595224287, −2.76184147861209524498395113499, −1.96985727032657027268391864894,
1.96985727032657027268391864894, 2.76184147861209524498395113499, 3.84928487291438368783595224287, 4.79922409943941392386911840550, 5.23210941933012756997585437601, 6.05763279934618195094641238203, 6.99940598940526418624129680332, 7.932734379283643434734382269335, 8.424769593532622569408273807934, 10.02944591943552765476781696696