| L(s) = 1 | − 5-s − 11-s − 13-s − 2·19-s + 2·23-s + 25-s + 10·29-s + 2·31-s + 2·41-s + 4·43-s + 4·47-s − 7·49-s + 55-s − 12·59-s + 14·61-s + 65-s − 4·73-s + 8·79-s − 4·83-s − 14·89-s + 2·95-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 0.277·13-s − 0.458·19-s + 0.417·23-s + 1/5·25-s + 1.85·29-s + 0.359·31-s + 0.312·41-s + 0.609·43-s + 0.583·47-s − 49-s + 0.134·55-s − 1.56·59-s + 1.79·61-s + 0.124·65-s − 0.468·73-s + 0.900·79-s − 0.439·83-s − 1.48·89-s + 0.205·95-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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
| \(L(1)\) |
\(\approx\) |
\(1.895061757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.895061757\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.73384952960069, −13.22153895594747, −12.65360400737035, −12.29438699897676, −11.85670234163227, −11.22998465300186, −10.80216496755167, −10.36573781322822, −9.759005951467086, −9.359673916602690, −8.470436222123115, −8.449317253286321, −7.735569018390228, −7.203899800357359, −6.688040758045134, −6.190616833711635, −5.548353009669757, −4.896340366098515, −4.473535871166608, −3.951111152641766, −3.100829709565619, −2.754498549279600, −2.021350316650281, −1.143492250284590, −0.4726409388088462,
0.4726409388088462, 1.143492250284590, 2.021350316650281, 2.754498549279600, 3.100829709565619, 3.951111152641766, 4.473535871166608, 4.896340366098515, 5.548353009669757, 6.190616833711635, 6.688040758045134, 7.203899800357359, 7.735569018390228, 8.449317253286321, 8.470436222123115, 9.359673916602690, 9.759005951467086, 10.36573781322822, 10.80216496755167, 11.22998465300186, 11.85670234163227, 12.29438699897676, 12.65360400737035, 13.22153895594747, 13.73384952960069
