| L(s) = 1 | + 5-s + 2·11-s − 13-s + 17-s + 8·19-s + 6·23-s + 25-s + 4·31-s + 2·37-s − 4·41-s + 4·43-s − 7·49-s + 6·53-s + 2·55-s + 4·59-s + 8·61-s − 65-s − 8·67-s − 4·71-s − 10·79-s + 8·83-s + 85-s − 6·89-s + 8·95-s + 8·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.603·11-s − 0.277·13-s + 0.242·17-s + 1.83·19-s + 1.25·23-s + 1/5·25-s + 0.718·31-s + 0.328·37-s − 0.624·41-s + 0.609·43-s − 49-s + 0.824·53-s + 0.269·55-s + 0.520·59-s + 1.02·61-s − 0.124·65-s − 0.977·67-s − 0.474·71-s − 1.12·79-s + 0.878·83-s + 0.108·85-s − 0.635·89-s + 0.820·95-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.044115457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.044115457\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.24390281427791, −12.97560852766587, −12.19347006539483, −11.89859998571864, −11.42649979324051, −11.00265529759397, −10.28790600616207, −9.876967516887839, −9.555074484822299, −8.946889758494944, −8.625070064865856, −7.880521511048954, −7.390156551365097, −6.973636241826169, −6.472570039833053, −5.806161570799204, −5.402833257312708, −4.860776236773374, −4.370346978852284, −3.557320280362559, −3.115117847643053, −2.607025876208210, −1.791572363500798, −1.141541433373856, −0.6638735151511538,
0.6638735151511538, 1.141541433373856, 1.791572363500798, 2.607025876208210, 3.115117847643053, 3.557320280362559, 4.370346978852284, 4.860776236773374, 5.402833257312708, 5.806161570799204, 6.472570039833053, 6.973636241826169, 7.390156551365097, 7.880521511048954, 8.625070064865856, 8.946889758494944, 9.555074484822299, 9.876967516887839, 10.28790600616207, 11.00265529759397, 11.42649979324051, 11.89859998571864, 12.19347006539483, 12.97560852766587, 13.24390281427791
