L(s) = 1 | + 1.73·2-s + 3-s + 1.99·4-s − 5-s + 1.73·6-s + 1.73·8-s + 9-s − 1.73·10-s + 1.99·12-s − 15-s + 0.999·16-s − 1.73·17-s + 1.73·18-s − 1.99·20-s − 23-s + 1.73·24-s + 25-s + 27-s − 1.73·30-s − 31-s − 2.99·34-s + 1.99·36-s − 1.73·40-s − 45-s − 1.73·46-s + 47-s + 0.999·48-s + ⋯ |
L(s) = 1 | + 1.73·2-s + 3-s + 1.99·4-s − 5-s + 1.73·6-s + 1.73·8-s + 9-s − 1.73·10-s + 1.99·12-s − 15-s + 0.999·16-s − 1.73·17-s + 1.73·18-s − 1.99·20-s − 23-s + 1.73·24-s + 25-s + 27-s − 1.73·30-s − 31-s − 2.99·34-s + 1.99·36-s − 1.73·40-s − 45-s − 1.73·46-s + 47-s + 0.999·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.449569442\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.449569442\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 1.73T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.73T + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.73T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.73T + T^{2} \) |
| 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
−9.264180899765024003070754522505, −8.570255334305873910561405012797, −7.61171311947797003367397914238, −7.03435184078224158050080459254, −6.23789872987372127702708394998, −5.07644779908904633346197681159, −4.16578495649104800449671578928, −3.90090377276499587792199817408, −2.84666865869189123637092785498, −2.03033035824921639371144693890,
2.03033035824921639371144693890, 2.84666865869189123637092785498, 3.90090377276499587792199817408, 4.16578495649104800449671578928, 5.07644779908904633346197681159, 6.23789872987372127702708394998, 7.03435184078224158050080459254, 7.61171311947797003367397914238, 8.570255334305873910561405012797, 9.264180899765024003070754522505