L(s) = 1 | − 3-s − 5-s − 2·9-s − 6·11-s + 2·13-s + 15-s − 6·17-s − 8·19-s − 3·23-s + 25-s + 5·27-s + 3·29-s − 2·31-s + 6·33-s + 8·37-s − 2·39-s − 3·41-s − 5·43-s + 2·45-s + 6·51-s + 12·53-s + 6·55-s + 8·57-s − 61-s − 2·65-s + 7·67-s + 3·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s − 1.80·11-s + 0.554·13-s + 0.258·15-s − 1.45·17-s − 1.83·19-s − 0.625·23-s + 1/5·25-s + 0.962·27-s + 0.557·29-s − 0.359·31-s + 1.04·33-s + 1.31·37-s − 0.320·39-s − 0.468·41-s − 0.762·43-s + 0.298·45-s + 0.840·51-s + 1.64·53-s + 0.809·55-s + 1.05·57-s − 0.128·61-s − 0.248·65-s + 0.855·67-s + 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4248641611\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4248641611\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.418457648261228800706744207254, −7.911720620171174460609980480330, −6.85800979915214965008662651215, −6.23959873569828731165287636724, −5.51833969623163735747186284172, −4.71220951123937270238193211276, −4.03899252884586194684861817620, −2.81211632700881415652661425899, −2.15452827007202200247945688539, −0.35800552561759987928458722720,
0.35800552561759987928458722720, 2.15452827007202200247945688539, 2.81211632700881415652661425899, 4.03899252884586194684861817620, 4.71220951123937270238193211276, 5.51833969623163735747186284172, 6.23959873569828731165287636724, 6.85800979915214965008662651215, 7.911720620171174460609980480330, 8.418457648261228800706744207254