| L(s) = 1 | + 3-s − 7-s − 2·9-s + 13-s − 4·19-s − 21-s − 5·27-s + 6·29-s + 31-s + 2·37-s + 39-s − 2·43-s + 6·47-s − 6·49-s − 3·53-s − 4·57-s − 6·59-s + 10·61-s + 2·63-s + 4·67-s + 3·71-s + 2·73-s + 79-s + 81-s − 12·83-s + 6·87-s − 6·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.277·13-s − 0.917·19-s − 0.218·21-s − 0.962·27-s + 1.11·29-s + 0.179·31-s + 0.328·37-s + 0.160·39-s − 0.304·43-s + 0.875·47-s − 6/7·49-s − 0.412·53-s − 0.529·57-s − 0.781·59-s + 1.28·61-s + 0.251·63-s + 0.488·67-s + 0.356·71-s + 0.234·73-s + 0.112·79-s + 1/9·81-s − 1.31·83-s + 0.643·87-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.870743333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.870743333\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−15.26530303097837, −14.50187892759277, −14.20630320021564, −13.67094271787825, −13.07123180057918, −12.64066447760423, −12.00317887960638, −11.41053436740471, −10.90620744652468, −10.29150955937782, −9.713444644874046, −9.160042083573306, −8.534339368858225, −8.237091579603191, −7.594951943940518, −6.756236727759150, −6.352345364211605, −5.710237521514620, −5.011375166188910, −4.249171266532877, −3.655247533659185, −2.909375697012957, −2.456992874934301, −1.571189509146740, −0.4998662878130997,
0.4998662878130997, 1.571189509146740, 2.456992874934301, 2.909375697012957, 3.655247533659185, 4.249171266532877, 5.011375166188910, 5.710237521514620, 6.352345364211605, 6.756236727759150, 7.594951943940518, 8.237091579603191, 8.534339368858225, 9.160042083573306, 9.713444644874046, 10.29150955937782, 10.90620744652468, 11.41053436740471, 12.00317887960638, 12.64066447760423, 13.07123180057918, 13.67094271787825, 14.20630320021564, 14.50187892759277, 15.26530303097837
