| L(s) = 1 | + 3-s − 2·7-s − 2·9-s − 5·11-s − 5·17-s − 5·19-s − 2·21-s + 6·23-s − 5·27-s + 4·29-s − 10·31-s − 5·33-s + 10·37-s + 5·41-s + 4·43-s − 8·47-s − 3·49-s − 5·51-s + 10·53-s − 5·57-s − 10·61-s + 4·63-s + 3·67-s + 6·69-s + 5·73-s + 10·77-s − 10·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s − 2/3·9-s − 1.50·11-s − 1.21·17-s − 1.14·19-s − 0.436·21-s + 1.25·23-s − 0.962·27-s + 0.742·29-s − 1.79·31-s − 0.870·33-s + 1.64·37-s + 0.780·41-s + 0.609·43-s − 1.16·47-s − 3/7·49-s − 0.700·51-s + 1.37·53-s − 0.662·57-s − 1.28·61-s + 0.503·63-s + 0.366·67-s + 0.722·69-s + 0.585·73-s + 1.13·77-s − 1.12·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−9.705858456866675900060541112818, −8.944564287296660537176265634374, −8.263311087506111478159685114153, −7.33621460188329495902519819950, −6.35555537184316601178413529953, −5.41791980271878332692465319450, −4.29436507895733982123025584023, −3.00596099350811580818924457861, −2.36178874928520003829114668262, 0,
2.36178874928520003829114668262, 3.00596099350811580818924457861, 4.29436507895733982123025584023, 5.41791980271878332692465319450, 6.35555537184316601178413529953, 7.33621460188329495902519819950, 8.263311087506111478159685114153, 8.944564287296660537176265634374, 9.705858456866675900060541112818
