| L(s) = 1 | + 3-s + 7-s + 9-s + 11-s + 2·13-s − 6·17-s − 2·19-s + 21-s + 8·23-s + 27-s − 8·29-s + 33-s − 6·37-s + 2·39-s + 8·41-s + 8·43-s + 6·47-s + 49-s − 6·51-s − 10·53-s − 2·57-s + 10·61-s + 63-s + 2·67-s + 8·69-s + 8·71-s + 6·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 1.45·17-s − 0.458·19-s + 0.218·21-s + 1.66·23-s + 0.192·27-s − 1.48·29-s + 0.174·33-s − 0.986·37-s + 0.320·39-s + 1.24·41-s + 1.21·43-s + 0.875·47-s + 1/7·49-s − 0.840·51-s − 1.37·53-s − 0.264·57-s + 1.28·61-s + 0.125·63-s + 0.244·67-s + 0.963·69-s + 0.949·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.176723573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.176723573\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.47714479437703, −14.23773417121853, −13.58184441975495, −12.98771334431846, −12.82276249478061, −12.11416711953360, −11.23493817147837, −11.00023693817101, −10.72912390896380, −9.734254779228649, −9.177603887765288, −8.946339794855879, −8.368229109642211, −7.759610481936288, −7.087996139946467, −6.757243742438973, −6.015567514792967, −5.355879725874651, −4.702217572951090, −4.067967547802444, −3.641420146238449, −2.748971695789493, −2.190904300504649, −1.493872735425377, −0.6190180101580109,
0.6190180101580109, 1.493872735425377, 2.190904300504649, 2.748971695789493, 3.641420146238449, 4.067967547802444, 4.702217572951090, 5.355879725874651, 6.015567514792967, 6.757243742438973, 7.087996139946467, 7.759610481936288, 8.368229109642211, 8.946339794855879, 9.177603887765288, 9.734254779228649, 10.72912390896380, 11.00023693817101, 11.23493817147837, 12.11416711953360, 12.82276249478061, 12.98771334431846, 13.58184441975495, 14.23773417121853, 14.47714479437703
