| L(s) = 1 | − 3-s − 3·7-s + 9-s + 11-s + 4·13-s + 17-s − 4·19-s + 3·21-s − 23-s − 27-s − 3·29-s + 5·31-s − 33-s + 6·37-s − 4·39-s + 8·41-s + 13·43-s − 12·47-s + 2·49-s − 51-s + 4·53-s + 4·57-s + 6·59-s − 10·61-s − 3·63-s + 8·67-s + 69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 0.242·17-s − 0.917·19-s + 0.654·21-s − 0.208·23-s − 0.192·27-s − 0.557·29-s + 0.898·31-s − 0.174·33-s + 0.986·37-s − 0.640·39-s + 1.24·41-s + 1.98·43-s − 1.75·47-s + 2/7·49-s − 0.140·51-s + 0.549·53-s + 0.529·57-s + 0.781·59-s − 1.28·61-s − 0.377·63-s + 0.977·67-s + 0.120·69-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
| \(L(1)\) |
\(\approx\) |
\(1.503353507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.503353507\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 13 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−12.89970503374009, −12.70823545361168, −11.99369646647266, −11.48563052223576, −11.18640515115054, −10.54859744826051, −10.30172875513914, −9.605918888709606, −9.319564342125402, −8.838969796082865, −8.153852219342751, −7.814806602394416, −7.072257232563063, −6.650546775346949, −6.075719594932253, −6.007641574757118, −5.382251736238372, −4.517158916482488, −4.138298059021035, −3.724637942198202, −2.979744728380673, −2.549631200309390, −1.684969968579059, −1.035605423296781, −0.4057499605293285,
0.4057499605293285, 1.035605423296781, 1.684969968579059, 2.549631200309390, 2.979744728380673, 3.724637942198202, 4.138298059021035, 4.517158916482488, 5.382251736238372, 6.007641574757118, 6.075719594932253, 6.650546775346949, 7.072257232563063, 7.814806602394416, 8.153852219342751, 8.838969796082865, 9.319564342125402, 9.605918888709606, 10.30172875513914, 10.54859744826051, 11.18640515115054, 11.48563052223576, 11.99369646647266, 12.70823545361168, 12.89970503374009
