| L(s) = 1 | − 3-s − 4·7-s + 9-s − 11-s − 2·13-s + 17-s + 2·19-s + 4·21-s + 4·23-s − 27-s + 6·29-s + 8·31-s + 33-s + 2·37-s + 2·39-s − 6·41-s + 10·43-s + 2·47-s + 9·49-s − 51-s − 10·53-s − 2·57-s − 10·59-s − 8·61-s − 4·63-s − 12·67-s − 4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 0.242·17-s + 0.458·19-s + 0.872·21-s + 0.834·23-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.174·33-s + 0.328·37-s + 0.320·39-s − 0.937·41-s + 1.52·43-s + 0.291·47-s + 9/7·49-s − 0.140·51-s − 1.37·53-s − 0.264·57-s − 1.30·59-s − 1.02·61-s − 0.503·63-s − 1.46·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.506025496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.506025496\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.85484588331229, −12.45228111313398, −12.03159363203018, −11.80465049221632, −10.95595101867441, −10.58039413230362, −10.18412240791386, −9.758523981000724, −9.202258492256808, −8.994208010522990, −8.137275670388732, −7.581556783690945, −7.294728068882586, −6.550952754309475, −6.236766746472613, −5.976721565593091, −5.085392075566100, −4.791574272531434, −4.263121137666378, −3.403082641156586, −3.023196877700565, −2.655446652266145, −1.756831154085024, −0.8856881978882928, −0.4505325257122909,
0.4505325257122909, 0.8856881978882928, 1.756831154085024, 2.655446652266145, 3.023196877700565, 3.403082641156586, 4.263121137666378, 4.791574272531434, 5.085392075566100, 5.976721565593091, 6.236766746472613, 6.550952754309475, 7.294728068882586, 7.581556783690945, 8.137275670388732, 8.994208010522990, 9.202258492256808, 9.758523981000724, 10.18412240791386, 10.58039413230362, 10.95595101867441, 11.80465049221632, 12.03159363203018, 12.45228111313398, 12.85484588331229
