| L(s) = 1 | + 7-s − 4·11-s + 6·13-s + 6·17-s + 4·19-s + 4·23-s + 2·29-s − 8·31-s − 6·37-s − 6·41-s + 8·43-s + 49-s + 6·53-s − 4·59-s + 10·61-s + 8·67-s + 12·71-s + 14·73-s − 4·77-s + 16·79-s + 12·83-s − 14·89-s + 6·91-s − 18·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.20·11-s + 1.66·13-s + 1.45·17-s + 0.917·19-s + 0.834·23-s + 0.371·29-s − 1.43·31-s − 0.986·37-s − 0.937·41-s + 1.21·43-s + 1/7·49-s + 0.824·53-s − 0.520·59-s + 1.28·61-s + 0.977·67-s + 1.42·71-s + 1.63·73-s − 0.455·77-s + 1.80·79-s + 1.31·83-s − 1.48·89-s + 0.628·91-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.135714153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.135714153\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.39269281790491, −13.93905460423378, −13.56904230519874, −13.04013768196184, −12.36890025836774, −12.13028653079255, −11.22231504266830, −10.86744778374887, −10.62155468027573, −9.754416158854022, −9.416966360417079, −8.624893668551294, −8.161900747483059, −7.814175246665298, −7.083215038098840, −6.632086626592514, −5.648285966529557, −5.401929820322323, −5.036576057774455, −3.886477535397541, −3.561814588224455, −2.920377274889504, −2.094384534738418, −1.262328250636514, −0.6933628801634937,
0.6933628801634937, 1.262328250636514, 2.094384534738418, 2.920377274889504, 3.561814588224455, 3.886477535397541, 5.036576057774455, 5.401929820322323, 5.648285966529557, 6.632086626592514, 7.083215038098840, 7.814175246665298, 8.161900747483059, 8.624893668551294, 9.416966360417079, 9.754416158854022, 10.62155468027573, 10.86744778374887, 11.22231504266830, 12.13028653079255, 12.36890025836774, 13.04013768196184, 13.56904230519874, 13.93905460423378, 14.39269281790491
