| L(s) = 1 | − 3-s + 9-s + 4·11-s − 6·13-s + 6·17-s − 8·19-s − 4·23-s − 27-s − 10·29-s − 4·33-s − 6·37-s + 6·39-s − 6·41-s − 43-s + 12·47-s − 7·49-s − 6·51-s − 6·53-s + 8·57-s − 4·59-s + 6·61-s − 4·67-s + 4·69-s − 8·71-s + 10·73-s + 8·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s − 1.66·13-s + 1.45·17-s − 1.83·19-s − 0.834·23-s − 0.192·27-s − 1.85·29-s − 0.696·33-s − 0.986·37-s + 0.960·39-s − 0.937·41-s − 0.152·43-s + 1.75·47-s − 49-s − 0.840·51-s − 0.824·53-s + 1.05·57-s − 0.520·59-s + 0.768·61-s − 0.488·67-s + 0.481·69-s − 0.949·71-s + 1.17·73-s + 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3762385296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3762385296\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−12.74087156455373, −12.51595784508110, −12.10090476618339, −11.77169992021944, −11.23745880576284, −10.56493247598951, −10.33910897880974, −9.713485091305128, −9.372916241415273, −8.913406553672329, −8.170051874410822, −7.767472587094167, −7.230375365064980, −6.790892436842508, −6.259647958078241, −5.817007518361335, −5.199737213065827, −4.834373719270822, −4.069606734064936, −3.800549551048469, −3.138405262679106, −2.188448540362457, −1.899858188340385, −1.175302063380381, −0.1820642115907189,
0.1820642115907189, 1.175302063380381, 1.899858188340385, 2.188448540362457, 3.138405262679106, 3.800549551048469, 4.069606734064936, 4.834373719270822, 5.199737213065827, 5.817007518361335, 6.259647958078241, 6.790892436842508, 7.230375365064980, 7.767472587094167, 8.170051874410822, 8.913406553672329, 9.372916241415273, 9.713485091305128, 10.33910897880974, 10.56493247598951, 11.23745880576284, 11.77169992021944, 12.10090476618339, 12.51595784508110, 12.74087156455373
