| L(s) = 1 | − 5-s + 4·7-s + 2·17-s + 4·19-s − 8·23-s + 25-s + 6·29-s − 4·35-s + 6·37-s + 10·41-s + 4·43-s − 8·47-s + 9·49-s − 10·53-s + 6·61-s − 4·67-s + 14·73-s − 16·79-s − 12·83-s − 2·85-s + 2·89-s − 4·95-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s + 0.485·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s + 1.11·29-s − 0.676·35-s + 0.986·37-s + 1.56·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s − 1.37·53-s + 0.768·61-s − 0.488·67-s + 1.63·73-s − 1.80·79-s − 1.31·83-s − 0.216·85-s + 0.211·89-s − 0.410·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.189522612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.189522612\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.85061049750495, −12.91795818206705, −12.58330527799078, −11.93514060618670, −11.65360312094097, −11.19540181420189, −10.82655600607903, −10.03264101354462, −9.813111034659190, −9.109431828033414, −8.463521074894345, −8.038242588235200, −7.748039865328823, −7.337784183311172, −6.529572850218843, −5.958836821806878, −5.474945629157462, −4.832996141987114, −4.403160946269178, −3.957751543409421, −3.167037107746077, −2.571507843427771, −1.842739651598640, −1.243555221793987, −0.5851363649947068,
0.5851363649947068, 1.243555221793987, 1.842739651598640, 2.571507843427771, 3.167037107746077, 3.957751543409421, 4.403160946269178, 4.832996141987114, 5.474945629157462, 5.958836821806878, 6.529572850218843, 7.337784183311172, 7.748039865328823, 8.038242588235200, 8.463521074894345, 9.109431828033414, 9.813111034659190, 10.03264101354462, 10.82655600607903, 11.19540181420189, 11.65360312094097, 11.93514060618670, 12.58330527799078, 12.91795818206705, 13.85061049750495
