L(s) = 1 | + 3-s − 2·7-s + 9-s − 6·11-s − 2·13-s + 2·17-s − 2·19-s − 2·21-s + 6·23-s + 27-s − 10·29-s + 4·31-s − 6·33-s + 2·37-s − 2·39-s + 6·41-s + 6·47-s − 3·49-s + 2·51-s − 10·53-s − 2·57-s + 6·59-s − 6·61-s − 2·63-s + 16·67-s + 6·69-s − 8·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.80·11-s − 0.554·13-s + 0.485·17-s − 0.458·19-s − 0.436·21-s + 1.25·23-s + 0.192·27-s − 1.85·29-s + 0.718·31-s − 1.04·33-s + 0.328·37-s − 0.320·39-s + 0.937·41-s + 0.875·47-s − 3/7·49-s + 0.280·51-s − 1.37·53-s − 0.264·57-s + 0.781·59-s − 0.768·61-s − 0.251·63-s + 1.95·67-s + 0.722·69-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.522551255\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.522551255\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.63673307457394675109969495567, −7.24363430315120775271601585969, −6.35394015861602454189251739401, −5.56052832606525336238042110966, −4.98943909657996553372240376787, −4.14780051953416484392191123229, −3.17542851103246843339167056831, −2.76260796484964071492101944422, −1.94880012443427660192473848218, −0.54105666062457426769681416030,
0.54105666062457426769681416030, 1.94880012443427660192473848218, 2.76260796484964071492101944422, 3.17542851103246843339167056831, 4.14780051953416484392191123229, 4.98943909657996553372240376787, 5.56052832606525336238042110966, 6.35394015861602454189251739401, 7.24363430315120775271601585969, 7.63673307457394675109969495567