L(s) = 1 | − 5-s + 4·7-s + 6·13-s − 2·17-s + 4·19-s + 8·23-s + 25-s + 6·29-s − 4·35-s − 2·37-s + 2·41-s + 8·43-s + 9·49-s + 2·53-s − 4·59-s + 10·61-s − 6·65-s + 12·67-s + 16·71-s − 6·73-s − 8·79-s − 16·83-s + 2·85-s − 10·89-s + 24·91-s − 4·95-s + 10·97-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.51·7-s + 1.66·13-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 − 0.328·37-s + 0.312·41-s + 1.21·43-s + 9/7·49-s + 0.274·53-s − 0.520·59-s + 1.28·61-s − 0.744·65-s + 1.46·67-s + 1.89·71-s − 0.702·73-s − 0.900·79-s − 1.75·83-s + 0.216·85-s − 1.05·89-s + 2.51·91-s − 0.410·95-s + 1.01·97-s + ⋯ |
Λ(s)=(=(43560s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(43560s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
3.996096929 |
L(21) |
≈ |
3.996096929 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 5 | 1+T |
| 11 | 1 |
good | 7 | 1−4T+pT2 |
| 13 | 1−6T+pT2 |
| 17 | 1+2T+pT2 |
| 19 | 1−4T+pT2 |
| 23 | 1−8T+pT2 |
| 29 | 1−6T+pT2 |
| 31 | 1+pT2 |
| 37 | 1+2T+pT2 |
| 41 | 1−2T+pT2 |
| 43 | 1−8T+pT2 |
| 47 | 1+pT2 |
| 53 | 1−2T+pT2 |
| 59 | 1+4T+pT2 |
| 61 | 1−10T+pT2 |
| 67 | 1−12T+pT2 |
| 71 | 1−16T+pT2 |
| 73 | 1+6T+pT2 |
| 79 | 1+8T+pT2 |
| 83 | 1+16T+pT2 |
| 89 | 1+10T+pT2 |
| 97 | 1−10T+pT2 |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−14.54148997017571, −14.14824706298236, −13.86839141659273, −13.01809629030151, −12.77780535692275, −11.89104653219775, −11.44191504672753, −11.11268676096999, −10.77162965235900, −10.07911576939431, −9.219054282201628, −8.766963751868313, −8.313754245616861, −7.923746829696269, −7.125696265970288, −6.808510780615954, −5.901465016287536, −5.385265244962661, −4.767158574160939, −4.273283151927510, −3.589731041560660, −2.923620460069711, −2.098670165146183, −1.151954464426075, −0.9016669122759157,
0.9016669122759157, 1.151954464426075, 2.098670165146183, 2.923620460069711, 3.589731041560660, 4.273283151927510, 4.767158574160939, 5.385265244962661, 5.901465016287536, 6.808510780615954, 7.125696265970288, 7.923746829696269, 8.313754245616861, 8.766963751868313, 9.219054282201628, 10.07911576939431, 10.77162965235900, 11.11268676096999, 11.44191504672753, 11.89104653219775, 12.77780535692275, 13.01809629030151, 13.86839141659273, 14.14824706298236, 14.54148997017571