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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.958716653\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.958716653\) |
\(L(1)\) |
|
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
−11.07067185158620858047463607831, −10.46569589200719895761701231757, −9.371246171645520668549116380100, −7.970835435007244695063973068796, −7.02666353442432348798601254878, −6.12269619030713511372617528053, −5.33569885850940032010161472943, −4.42938801666431015361494420647, −3.20497650277098718036239044932, −2.52899724031951772135360380788,
2.52899724031951772135360380788, 3.20497650277098718036239044932, 4.42938801666431015361494420647, 5.33569885850940032010161472943, 6.12269619030713511372617528053, 7.02666353442432348798601254878, 7.970835435007244695063973068796, 9.371246171645520668549116380100, 10.46569589200719895761701231757, 11.07067185158620858047463607831