| L(s) = 1 | − 5-s + 3·11-s + 4·13-s + 6·17-s + 8·19-s − 6·23-s + 25-s − 3·29-s + 5·31-s + 2·37-s − 6·41-s + 10·43-s − 12·47-s + 6·53-s − 3·55-s − 9·59-s − 14·61-s − 4·65-s + 4·67-s − 6·71-s + 13·73-s + 79-s + 3·83-s − 6·85-s − 12·89-s − 8·95-s − 5·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.904·11-s + 1.10·13-s + 1.45·17-s + 1.83·19-s − 1.25·23-s + 1/5·25-s − 0.557·29-s + 0.898·31-s + 0.328·37-s − 0.937·41-s + 1.52·43-s − 1.75·47-s + 0.824·53-s − 0.404·55-s − 1.17·59-s − 1.79·61-s − 0.496·65-s + 0.488·67-s − 0.712·71-s + 1.52·73-s + 0.112·79-s + 0.329·83-s − 0.650·85-s − 1.27·89-s − 0.820·95-s − 0.507·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.125064946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.125064946\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−13.86301301442173, −13.33822209909454, −12.52676600969675, −12.19049060473054, −11.75211918890235, −11.38600071404333, −10.87265665616228, −10.11496982436975, −9.819131204807603, −9.276925673007702, −8.778881607234036, −8.065385975683035, −7.776080448903549, −7.340793011062974, −6.517107692472592, −6.154610084840600, −5.587885846746746, −5.063077649298450, −4.288051505095261, −3.773319508493406, −3.324182971600686, −2.845385096237666, −1.693103925015806, −1.274510465488229, −0.6075786804734609,
0.6075786804734609, 1.274510465488229, 1.693103925015806, 2.845385096237666, 3.324182971600686, 3.773319508493406, 4.288051505095261, 5.063077649298450, 5.587885846746746, 6.154610084840600, 6.517107692472592, 7.340793011062974, 7.776080448903549, 8.065385975683035, 8.778881607234036, 9.276925673007702, 9.819131204807603, 10.11496982436975, 10.87265665616228, 11.38600071404333, 11.75211918890235, 12.19049060473054, 12.52676600969675, 13.33822209909454, 13.86301301442173
