| L(s) = 1 | + 5-s + 2·7-s + 11-s − 13-s + 2·17-s + 6·23-s + 25-s − 4·29-s + 4·31-s + 2·35-s − 8·37-s − 6·41-s − 4·43-s − 12·47-s − 3·49-s − 2·53-s + 55-s − 4·59-s − 2·61-s − 65-s − 2·67-s + 8·71-s + 4·73-s + 2·77-s + 4·79-s − 4·83-s + 2·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s + 0.301·11-s − 0.277·13-s + 0.485·17-s + 1.25·23-s + 1/5·25-s − 0.742·29-s + 0.718·31-s + 0.338·35-s − 1.31·37-s − 0.937·41-s − 0.609·43-s − 1.75·47-s − 3/7·49-s − 0.274·53-s + 0.134·55-s − 0.520·59-s − 0.256·61-s − 0.124·65-s − 0.244·67-s + 0.949·71-s + 0.468·73-s + 0.227·77-s + 0.450·79-s − 0.439·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.802959417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.802959417\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.68997918528138, −13.31776144516268, −12.77380751559041, −12.23012279987044, −11.75405703746765, −11.29841099151296, −10.80815835332659, −10.29816902825272, −9.752950760518281, −9.328035684505753, −8.750375271366409, −8.247798582273025, −7.795699267084695, −7.160233905805283, −6.627759518231420, −6.240027185665563, −5.325634862717702, −5.081718875682162, −4.658774689842374, −3.752293820300670, −3.262428501419646, −2.641742386750021, −1.663925411994410, −1.559270835748067, −0.5145874247770196,
0.5145874247770196, 1.559270835748067, 1.663925411994410, 2.641742386750021, 3.262428501419646, 3.752293820300670, 4.658774689842374, 5.081718875682162, 5.325634862717702, 6.240027185665563, 6.627759518231420, 7.160233905805283, 7.795699267084695, 8.247798582273025, 8.750375271366409, 9.328035684505753, 9.752950760518281, 10.29816902825272, 10.80815835332659, 11.29841099151296, 11.75405703746765, 12.23012279987044, 12.77380751559041, 13.31776144516268, 13.68997918528138
