L(s) = 1 | − 3-s − 3.73·5-s − 0.404·7-s + 9-s − 11-s + 13-s + 3.73·15-s − 1.32·17-s − 0.895·19-s + 0.404·21-s + 2.40·23-s + 8.92·25-s − 27-s − 8.14·29-s − 5.23·31-s + 33-s + 1.50·35-s − 0.808·37-s − 39-s + 1.59·41-s − 7.64·43-s − 3.73·45-s + 6.32·47-s − 6.83·49-s + 1.32·51-s + 5.51·53-s + 3.73·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.66·5-s − 0.152·7-s + 0.333·9-s − 0.301·11-s + 0.277·13-s + 0.963·15-s − 0.321·17-s − 0.205·19-s + 0.0882·21-s + 0.501·23-s + 1.78·25-s − 0.192·27-s − 1.51·29-s − 0.941·31-s + 0.174·33-s + 0.254·35-s − 0.132·37-s − 0.160·39-s + 0.249·41-s − 1.16·43-s − 0.556·45-s + 0.923·47-s − 0.976·49-s + 0.185·51-s + 0.758·53-s + 0.503·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4570595534\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4570595534\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + 3.73T + 5T^{2} \) |
| 7 | \( 1 + 0.404T + 7T^{2} \) |
| 17 | \( 1 + 1.32T + 17T^{2} \) |
| 19 | \( 1 + 0.895T + 19T^{2} \) |
| 23 | \( 1 - 2.40T + 23T^{2} \) |
| 29 | \( 1 + 8.14T + 29T^{2} \) |
| 31 | \( 1 + 5.23T + 31T^{2} \) |
| 37 | \( 1 + 0.808T + 37T^{2} \) |
| 41 | \( 1 - 1.59T + 41T^{2} \) |
| 43 | \( 1 + 7.64T + 43T^{2} \) |
| 47 | \( 1 - 6.32T + 47T^{2} \) |
| 53 | \( 1 - 5.51T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 - 6.97T + 61T^{2} \) |
| 67 | \( 1 + 5.73T + 67T^{2} \) |
| 71 | \( 1 + 15.9T + 71T^{2} \) |
| 73 | \( 1 + 1.92T + 73T^{2} \) |
| 79 | \( 1 - 2.40T + 79T^{2} \) |
| 83 | \( 1 + 7.46T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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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
−7.72185965013121304755333036187, −7.39191605417648692965790939041, −6.69786853869800457853480737998, −5.82406941178486761749977633808, −5.06175522420810728079885182099, −4.29889896091032569546861571460, −3.71710814530913938887336401668, −2.96376126291767695773861200308, −1.64253520715635596788854387636, −0.35521481485018965449200325244,
0.35521481485018965449200325244, 1.64253520715635596788854387636, 2.96376126291767695773861200308, 3.71710814530913938887336401668, 4.29889896091032569546861571460, 5.06175522420810728079885182099, 5.82406941178486761749977633808, 6.69786853869800457853480737998, 7.39191605417648692965790939041, 7.72185965013121304755333036187