| L(s) = 1 | − 5-s − 11-s + 4·13-s − 2·17-s − 2·19-s + 8·23-s + 25-s + 4·29-s + 4·31-s − 2·37-s − 12·41-s + 8·43-s − 4·47-s + 10·53-s + 55-s + 4·59-s − 6·61-s − 4·65-s + 12·67-s + 16·73-s + 6·79-s − 2·83-s + 2·85-s + 18·89-s + 2·95-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s + 1.10·13-s − 0.485·17-s − 0.458·19-s + 1.66·23-s + 1/5·25-s + 0.742·29-s + 0.718·31-s − 0.328·37-s − 1.87·41-s + 1.21·43-s − 0.583·47-s + 1.37·53-s + 0.134·55-s + 0.520·59-s − 0.768·61-s − 0.496·65-s + 1.46·67-s + 1.87·73-s + 0.675·79-s − 0.219·83-s + 0.216·85-s + 1.90·89-s + 0.205·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.595994376\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.595994376\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−13.66986587145670, −13.37283106202408, −12.80750378117518, −12.37730534936206, −11.77129516849756, −11.28152591303471, −10.88128165480587, −10.44700155338234, −9.913146382848898, −9.190276371101752, −8.639580955859101, −8.474059764276821, −7.820601011090606, −7.173037203397457, −6.649278674691716, −6.326110654955379, −5.555969654562731, −4.889235252125059, −4.622862357216666, −3.659113015250067, −3.472852797278404, −2.629917679539284, −2.044433976432526, −1.093816424671608, −0.5883905977552379,
0.5883905977552379, 1.093816424671608, 2.044433976432526, 2.629917679539284, 3.472852797278404, 3.659113015250067, 4.622862357216666, 4.889235252125059, 5.555969654562731, 6.326110654955379, 6.649278674691716, 7.173037203397457, 7.820601011090606, 8.474059764276821, 8.639580955859101, 9.190276371101752, 9.913146382848898, 10.44700155338234, 10.88128165480587, 11.28152591303471, 11.77129516849756, 12.37730534936206, 12.80750378117518, 13.37283106202408, 13.66986587145670
