| L(s) = 1 | + 2·5-s + 6·11-s + 3·13-s + 4·17-s + 5·19-s + 4·23-s − 25-s + 4·29-s − 7·31-s − 9·37-s − 2·41-s − 43-s + 2·47-s − 8·53-s + 12·55-s − 10·61-s + 6·65-s − 15·67-s + 6·71-s + 11·73-s + 79-s + 6·83-s + 8·85-s − 8·89-s + 10·95-s + 14·97-s − 6·101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.80·11-s + 0.832·13-s + 0.970·17-s + 1.14·19-s + 0.834·23-s − 1/5·25-s + 0.742·29-s − 1.25·31-s − 1.47·37-s − 0.312·41-s − 0.152·43-s + 0.291·47-s − 1.09·53-s + 1.61·55-s − 1.28·61-s + 0.744·65-s − 1.83·67-s + 0.712·71-s + 1.28·73-s + 0.112·79-s + 0.658·83-s + 0.867·85-s − 0.847·89-s + 1.02·95-s + 1.42·97-s − 0.597·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.886069730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.886069730\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.864958635697251334853880647464, −7.78948737661891823363458382576, −6.96985484211089758244572772534, −6.29947716155187759683682250311, −5.66387302190222474541069424081, −4.87818728357530028484620824477, −3.71276911737092808329334510131, −3.20368264259735866796427604516, −1.72733685010788976096784023703, −1.15804168709393057407778390929,
1.15804168709393057407778390929, 1.72733685010788976096784023703, 3.20368264259735866796427604516, 3.71276911737092808329334510131, 4.87818728357530028484620824477, 5.66387302190222474541069424081, 6.29947716155187759683682250311, 6.96985484211089758244572772534, 7.78948737661891823363458382576, 8.864958635697251334853880647464
