| L(s) = 1 | + 3-s + 7-s + 9-s − 4·11-s − 6·13-s − 6·17-s − 4·19-s + 21-s − 4·23-s + 27-s + 2·29-s − 8·31-s − 4·33-s + 6·37-s − 6·39-s + 6·41-s − 8·43-s + 49-s − 6·51-s + 6·53-s − 4·57-s − 4·59-s − 10·61-s + 63-s − 8·67-s − 4·69-s − 12·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 1.45·17-s − 0.917·19-s + 0.218·21-s − 0.834·23-s + 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.696·33-s + 0.986·37-s − 0.960·39-s + 0.937·41-s − 1.21·43-s + 1/7·49-s − 0.840·51-s + 0.824·53-s − 0.529·57-s − 0.520·59-s − 1.28·61-s + 0.125·63-s − 0.977·67-s − 0.481·69-s − 1.42·71-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
| \(L(1)\) |
\(\approx\) |
\(0.8411504669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8411504669\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.97884459283951, −14.74835428552927, −13.81677607032561, −13.48273872896332, −12.98927894418611, −12.30055617684004, −12.09246662384388, −11.05313777467708, −10.78193460160634, −10.24060749238715, −9.504138249267760, −9.190450025154591, −8.413699389348352, −7.927330765386029, −7.506086791463633, −6.900222184755234, −6.228264383357504, −5.466589888864204, −4.749365157365002, −4.476378778753983, −3.661450814029851, −2.658031259877006, −2.348227251928640, −1.767205077180602, −0.3039049614867913,
0.3039049614867913, 1.767205077180602, 2.348227251928640, 2.658031259877006, 3.661450814029851, 4.476378778753983, 4.749365157365002, 5.466589888864204, 6.228264383357504, 6.900222184755234, 7.506086791463633, 7.927330765386029, 8.413699389348352, 9.190450025154591, 9.504138249267760, 10.24060749238715, 10.78193460160634, 11.05313777467708, 12.09246662384388, 12.30055617684004, 12.98927894418611, 13.48273872896332, 13.81677607032561, 14.74835428552927, 14.97884459283951
