L(s) = 1 | − 3-s + 7-s + 9-s + 2·13-s + 6·17-s + 8·19-s − 21-s − 27-s − 6·29-s + 4·31-s − 10·37-s − 2·39-s − 6·41-s + 4·43-s + 49-s − 6·51-s − 6·53-s − 8·57-s − 12·59-s + 10·61-s + 63-s + 4·67-s − 12·71-s + 10·73-s − 8·79-s + 81-s − 12·83-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.554·13-s + 1.45·17-s + 1.83·19-s − 0.218·21-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 1.64·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s + 1/7·49-s − 0.840·51-s − 0.824·53-s − 1.05·57-s − 1.56·59-s + 1.28·61-s + 0.125·63-s + 0.488·67-s − 1.42·71-s + 1.17·73-s − 0.900·79-s + 1/9·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.38173125662640, −14.69417931910870, −14.06601430214938, −13.84791189574133, −13.15024796483524, −12.48976979488483, −12.02603343785332, −11.60029367457718, −11.12192108593823, −10.46546655730752, −9.952652722826528, −9.504499210949518, −8.805964258451988, −8.145602217510334, −7.558937414192017, −7.190353622651388, −6.407051253562484, −5.728339939339206, −5.299309493119517, −4.869365141038083, −3.842884400223550, −3.443398497956361, −2.676336110937512, −1.446296991014833, −1.231914778501832, 0,
1.231914778501832, 1.446296991014833, 2.676336110937512, 3.443398497956361, 3.842884400223550, 4.869365141038083, 5.299309493119517, 5.728339939339206, 6.407051253562484, 7.190353622651388, 7.558937414192017, 8.145602217510334, 8.805964258451988, 9.504499210949518, 9.952652722826528, 10.46546655730752, 11.12192108593823, 11.60029367457718, 12.02603343785332, 12.48976979488483, 13.15024796483524, 13.84791189574133, 14.06601430214938, 14.69417931910870, 15.38173125662640