| L(s) = 1 | − 5-s − 2·7-s − 4·13-s + 2·17-s − 4·19-s + 8·23-s + 25-s + 10·29-s + 4·31-s + 2·35-s + 8·43-s + 8·47-s − 3·49-s + 6·53-s − 14·59-s + 14·61-s + 4·65-s − 4·67-s + 12·71-s − 6·73-s + 12·79-s − 4·83-s − 2·85-s − 12·89-s + 8·91-s + 4·95-s − 14·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 1.10·13-s + 0.485·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s + 1.85·29-s + 0.718·31-s + 0.338·35-s + 1.21·43-s + 1.16·47-s − 3/7·49-s + 0.824·53-s − 1.82·59-s + 1.79·61-s + 0.496·65-s − 0.488·67-s + 1.42·71-s − 0.702·73-s + 1.35·79-s − 0.439·83-s − 0.216·85-s − 1.27·89-s + 0.838·91-s + 0.410·95-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.638745913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.638745913\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−14.77673629652145, −14.09382512039963, −13.78172728869151, −12.94411432118055, −12.56551597906348, −12.24241403428156, −11.68764995444602, −10.88073380762538, −10.61578461183659, −9.871356497415616, −9.554031337003252, −8.806953517652706, −8.397875650648143, −7.712197531332753, −7.096978403402788, −6.727245503885016, −6.121213850693776, −5.361304738965803, −4.745247664038095, −4.272596544122096, −3.496856389019430, −2.719707836846295, −2.515292174271973, −1.195413486817241, −0.5091432949619300,
0.5091432949619300, 1.195413486817241, 2.515292174271973, 2.719707836846295, 3.496856389019430, 4.272596544122096, 4.745247664038095, 5.361304738965803, 6.121213850693776, 6.727245503885016, 7.096978403402788, 7.712197531332753, 8.397875650648143, 8.806953517652706, 9.554031337003252, 9.871356497415616, 10.61578461183659, 10.88073380762538, 11.68764995444602, 12.24241403428156, 12.56551597906348, 12.94411432118055, 13.78172728869151, 14.09382512039963, 14.77673629652145
