| L(s) = 1 | + 3-s − 4·7-s + 9-s + 3·11-s + 7·13-s + 5·17-s + 6·19-s − 4·21-s − 4·23-s + 27-s + 2·29-s − 4·31-s + 3·33-s + 2·37-s + 7·39-s + 6·41-s − 43-s − 3·47-s + 9·49-s + 5·51-s + 6·53-s + 6·57-s + 3·59-s − 4·63-s + 3·67-s − 4·69-s − 6·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.904·11-s + 1.94·13-s + 1.21·17-s + 1.37·19-s − 0.872·21-s − 0.834·23-s + 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.522·33-s + 0.328·37-s + 1.12·39-s + 0.937·41-s − 0.152·43-s − 0.437·47-s + 9/7·49-s + 0.700·51-s + 0.824·53-s + 0.794·57-s + 0.390·59-s − 0.503·63-s + 0.366·67-s − 0.481·69-s − 0.712·71-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
| \(L(1)\) |
\(\approx\) |
\(4.162088244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.162088244\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−13.18768668507451, −12.67527066964262, −12.04538371656047, −11.77928859372902, −11.19550481803537, −10.58748388836177, −10.08012366824683, −9.648518076064397, −9.354828292195776, −8.829103219242009, −8.364510560369793, −7.838782787697953, −7.249883072311331, −6.802378495654150, −6.281701876777235, −5.706346991713175, −5.629656816315998, −4.496581335608483, −3.876525172402827, −3.583727471755805, −3.194104029637071, −2.665013636822929, −1.690190217753649, −1.167376712517239, −0.6156260951192716,
0.6156260951192716, 1.167376712517239, 1.690190217753649, 2.665013636822929, 3.194104029637071, 3.583727471755805, 3.876525172402827, 4.496581335608483, 5.629656816315998, 5.706346991713175, 6.281701876777235, 6.802378495654150, 7.249883072311331, 7.838782787697953, 8.364510560369793, 8.829103219242009, 9.354828292195776, 9.648518076064397, 10.08012366824683, 10.58748388836177, 11.19550481803537, 11.77928859372902, 12.04538371656047, 12.67527066964262, 13.18768668507451
