L(s) = 1 | + 2·5-s − 7-s − 2·11-s + 6·17-s − 4·19-s + 2·23-s − 25-s + 4·31-s − 2·35-s − 2·37-s + 2·41-s − 4·43-s + 49-s + 12·53-s − 4·55-s + 6·61-s + 12·67-s − 2·71-s + 2·73-s + 2·77-s − 4·83-s + 12·85-s + 2·89-s − 8·95-s + 10·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.377·7-s − 0.603·11-s + 1.45·17-s − 0.917·19-s + 0.417·23-s − 1/5·25-s + 0.718·31-s − 0.338·35-s − 0.328·37-s + 0.312·41-s − 0.609·43-s + 1/7·49-s + 1.64·53-s − 0.539·55-s + 0.768·61-s + 1.46·67-s − 0.237·71-s + 0.234·73-s + 0.227·77-s − 0.439·83-s + 1.30·85-s + 0.211·89-s − 0.820·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.730766231\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.730766231\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.80954948404778, −13.50298997328760, −12.84627725194733, −12.67668662081551, −11.95230666274696, −11.52866649631808, −10.80879835041526, −10.29123038850320, −9.955625395585152, −9.647525505597639, −8.802629600937055, −8.524788190152307, −7.800184874843119, −7.353288646133574, −6.658994871417933, −6.201646934427014, −5.634474275557255, −5.245343479013229, −4.626075925621732, −3.796666751873473, −3.329249106349979, −2.521552696753869, −2.153698526143100, −1.275163399909074, −0.5509073838096738,
0.5509073838096738, 1.275163399909074, 2.153698526143100, 2.521552696753869, 3.329249106349979, 3.796666751873473, 4.626075925621732, 5.245343479013229, 5.634474275557255, 6.201646934427014, 6.658994871417933, 7.353288646133574, 7.800184874843119, 8.524788190152307, 8.802629600937055, 9.647525505597639, 9.955625395585152, 10.29123038850320, 10.80879835041526, 11.52866649631808, 11.95230666274696, 12.67668662081551, 12.84627725194733, 13.50298997328760, 13.80954948404778