| L(s) = 1 | − 3·9-s + 11-s + 2·13-s − 2·17-s − 4·19-s − 8·23-s − 2·29-s − 4·31-s − 6·37-s − 6·41-s − 4·43-s + 12·47-s + 10·53-s + 8·59-s − 10·61-s + 4·67-s + 8·71-s − 2·73-s + 8·79-s + 9·81-s − 12·83-s − 10·89-s + 10·97-s − 3·99-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 9-s + 0.301·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s − 1.66·23-s − 0.371·29-s − 0.718·31-s − 0.986·37-s − 0.937·41-s − 0.609·43-s + 1.75·47-s + 1.37·53-s + 1.04·59-s − 1.28·61-s + 0.488·67-s + 0.949·71-s − 0.234·73-s + 0.900·79-s + 81-s − 1.31·83-s − 1.05·89-s + 1.01·97-s − 0.301·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−13.31773660442914, −12.73725168510355, −12.15901424404047, −11.90866925283002, −11.30998119210764, −10.99338319953260, −10.35498902901791, −10.12851388598536, −9.368415410627940, −8.852595432261435, −8.546496072063707, −8.218370019922677, −7.479436784599902, −7.038116648634233, −6.374390514732943, −6.068876570394360, −5.529011322299152, −5.082610098977956, −4.286270150577294, −3.827474881977762, −3.481204935914101, −2.638823398406406, −2.093225073880803, −1.676039537517895, −0.6395461924406758, 0,
0.6395461924406758, 1.676039537517895, 2.093225073880803, 2.638823398406406, 3.481204935914101, 3.827474881977762, 4.286270150577294, 5.082610098977956, 5.529011322299152, 6.068876570394360, 6.374390514732943, 7.038116648634233, 7.479436784599902, 8.218370019922677, 8.546496072063707, 8.852595432261435, 9.368415410627940, 10.12851388598536, 10.35498902901791, 10.99338319953260, 11.30998119210764, 11.90866925283002, 12.15901424404047, 12.73725168510355, 13.31773660442914
