| L(s) = 1 | − 3-s − 5·7-s + 9-s − 11-s + 6·13-s − 17-s + 5·21-s − 4·23-s − 27-s − 29-s − 4·31-s + 33-s + 2·37-s − 6·39-s − 9·41-s − 6·43-s + 3·47-s + 18·49-s + 51-s − 6·53-s + 7·59-s − 5·61-s − 5·63-s + 5·67-s + 4·69-s − 10·71-s + 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.88·7-s + 1/3·9-s − 0.301·11-s + 1.66·13-s − 0.242·17-s + 1.09·21-s − 0.834·23-s − 0.192·27-s − 0.185·29-s − 0.718·31-s + 0.174·33-s + 0.328·37-s − 0.960·39-s − 1.40·41-s − 0.914·43-s + 0.437·47-s + 18/7·49-s + 0.140·51-s − 0.824·53-s + 0.911·59-s − 0.640·61-s − 0.629·63-s + 0.610·67-s + 0.481·69-s − 1.18·71-s + 0.234·73-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.18333296698281, −13.04720583107837, −12.61216619941784, −11.95157694986205, −11.69487389509542, −10.85602537362021, −10.81017845824199, −10.07187123398176, −9.786706569465568, −9.335655301218089, −8.659469162180185, −8.407821173708198, −7.687464016271679, −6.977177531264005, −6.724767876567034, −6.227530315531962, −5.780579564919910, −5.483971904663183, −4.655860550089408, −3.894897244513069, −3.722882233490334, −3.104219811538527, −2.542672222685640, −1.673257625789490, −1.104436952580190, 0, 0,
1.104436952580190, 1.673257625789490, 2.542672222685640, 3.104219811538527, 3.722882233490334, 3.894897244513069, 4.655860550089408, 5.483971904663183, 5.780579564919910, 6.227530315531962, 6.724767876567034, 6.977177531264005, 7.687464016271679, 8.407821173708198, 8.659469162180185, 9.335655301218089, 9.786706569465568, 10.07187123398176, 10.81017845824199, 10.85602537362021, 11.69487389509542, 11.95157694986205, 12.61216619941784, 13.04720583107837, 13.18333296698281
