L(s) = 1 | + 1.41·5-s + 4·7-s + 5.65·11-s − 4.24·13-s + 6·17-s + 5.65·19-s − 8·23-s − 2.99·25-s − 4.24·29-s + 4·31-s + 5.65·35-s − 1.41·37-s + 2·41-s − 5.65·43-s + 8·47-s + 9·49-s + 9.89·53-s + 8.00·55-s − 4.24·61-s − 6·65-s + 11.3·67-s + 10·73-s + 22.6·77-s + 12·79-s + 5.65·83-s + 8.48·85-s − 16·89-s + ⋯ |
L(s) = 1 | + 0.632·5-s + 1.51·7-s + 1.70·11-s − 1.17·13-s + 1.45·17-s + 1.29·19-s − 1.66·23-s − 0.599·25-s − 0.787·29-s + 0.718·31-s + 0.956·35-s − 0.232·37-s + 0.312·41-s − 0.862·43-s + 1.16·47-s + 1.28·49-s + 1.35·53-s + 1.07·55-s − 0.543·61-s − 0.744·65-s + 1.38·67-s + 1.17·73-s + 2.57·77-s + 1.35·79-s + 0.620·83-s + 0.920·85-s − 1.69·89-s + ⋯ |
Λ(s)=(=(9216s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(9216s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
3.491933416 |
L(21) |
≈ |
3.491933416 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
good | 5 | 1−1.41T+5T2 |
| 7 | 1−4T+7T2 |
| 11 | 1−5.65T+11T2 |
| 13 | 1+4.24T+13T2 |
| 17 | 1−6T+17T2 |
| 19 | 1−5.65T+19T2 |
| 23 | 1+8T+23T2 |
| 29 | 1+4.24T+29T2 |
| 31 | 1−4T+31T2 |
| 37 | 1+1.41T+37T2 |
| 41 | 1−2T+41T2 |
| 43 | 1+5.65T+43T2 |
| 47 | 1−8T+47T2 |
| 53 | 1−9.89T+53T2 |
| 59 | 1+59T2 |
| 61 | 1+4.24T+61T2 |
| 67 | 1−11.3T+67T2 |
| 71 | 1+71T2 |
| 73 | 1−10T+73T2 |
| 79 | 1−12T+79T2 |
| 83 | 1−5.65T+83T2 |
| 89 | 1+16T+89T2 |
| 97 | 1−8T+97T2 |
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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
−7.73536282952643018202134004908, −7.17896038249826618637825140691, −6.27475982917468646684113643983, −5.51064588113498983081165322552, −5.14928741387013799069560225998, −4.16596500324326316740245778538, −3.62793282621066475013107164267, −2.37921804647973866560352906238, −1.69213857367136046510671702177, −0.988736452349166151317044692196,
0.988736452349166151317044692196, 1.69213857367136046510671702177, 2.37921804647973866560352906238, 3.62793282621066475013107164267, 4.16596500324326316740245778538, 5.14928741387013799069560225998, 5.51064588113498983081165322552, 6.27475982917468646684113643983, 7.17896038249826618637825140691, 7.73536282952643018202134004908