L(s) = 1 | − 1.73·3-s + 5-s + 1.99·9-s − 1.73·15-s + 1.73·23-s + 25-s − 1.73·27-s − 29-s − 41-s + 1.73·43-s + 1.99·45-s − 61-s − 1.73·67-s − 2.99·69-s − 1.73·75-s + 0.999·81-s + 1.73·83-s + 1.73·87-s + 89-s + 101-s + 1.73·103-s − 1.73·107-s + 109-s + 1.73·115-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 5-s + 1.99·9-s − 1.73·15-s + 1.73·23-s + 25-s − 1.73·27-s − 29-s − 41-s + 1.73·43-s + 1.99·45-s − 61-s − 1.73·67-s − 2.99·69-s − 1.73·75-s + 0.999·81-s + 1.73·83-s + 1.73·87-s + 89-s + 101-s + 1.73·103-s − 1.73·107-s + 109-s + 1.73·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8958430096\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8958430096\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 1.73T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.73T + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - 1.73T + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + 1.73T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 1.73T + T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 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
−8.956108210562206209990784044381, −7.59514839199470243006805898537, −6.96165348348636394809829758425, −6.26634060818155616380010169127, −5.70779353000963306177534561303, −5.07946398376705954394716485155, −4.46792955475968031660661767907, −3.18649694421326464983934446220, −1.91424914110462113324793165761, −0.915010044279353065802582012378,
0.915010044279353065802582012378, 1.91424914110462113324793165761, 3.18649694421326464983934446220, 4.46792955475968031660661767907, 5.07946398376705954394716485155, 5.70779353000963306177534561303, 6.26634060818155616380010169127, 6.96165348348636394809829758425, 7.59514839199470243006805898537, 8.956108210562206209990784044381