L(s) = 1 | + 0.414·3-s − 2.82·5-s + 1.58·7-s − 2.82·9-s − 5.24·11-s − 1.17·15-s − 0.171·17-s + 7.24·19-s + 0.656·21-s − 7.24·23-s + 3.00·25-s − 2.41·27-s − 2.65·29-s − 5.65·31-s − 2.17·33-s − 4.48·35-s + 9.48·37-s − 0.171·41-s − 10.0·43-s + 8.00·45-s + 6·47-s − 4.48·49-s − 0.0710·51-s + 2.82·53-s + 14.8·55-s + 2.99·57-s + 7.24·59-s + ⋯ |
L(s) = 1 | + 0.239·3-s − 1.26·5-s + 0.599·7-s − 0.942·9-s − 1.58·11-s − 0.302·15-s − 0.0416·17-s + 1.66·19-s + 0.143·21-s − 1.51·23-s + 0.600·25-s − 0.464·27-s − 0.493·29-s − 1.01·31-s − 0.378·33-s − 0.758·35-s + 1.55·37-s − 0.0267·41-s − 1.53·43-s + 1.19·45-s + 0.875·47-s − 0.640·49-s − 0.00995·51-s + 0.388·53-s + 1.99·55-s + 0.397·57-s + 0.942·59-s + ⋯ |
Λ(s)=(=(5408s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(5408s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
0.8980517640 |
L(21) |
≈ |
0.8980517640 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 13 | 1 |
good | 3 | 1−0.414T+3T2 |
| 5 | 1+2.82T+5T2 |
| 7 | 1−1.58T+7T2 |
| 11 | 1+5.24T+11T2 |
| 17 | 1+0.171T+17T2 |
| 19 | 1−7.24T+19T2 |
| 23 | 1+7.24T+23T2 |
| 29 | 1+2.65T+29T2 |
| 31 | 1+5.65T+31T2 |
| 37 | 1−9.48T+37T2 |
| 41 | 1+0.171T+41T2 |
| 43 | 1+10.0T+43T2 |
| 47 | 1−6T+47T2 |
| 53 | 1−2.82T+53T2 |
| 59 | 1−7.24T+59T2 |
| 61 | 1−7T+61T2 |
| 67 | 1−4.75T+67T2 |
| 71 | 1−1.24T+71T2 |
| 73 | 1+4.48T+73T2 |
| 79 | 1+6T+79T2 |
| 83 | 1−4T+83T2 |
| 89 | 1+14.6T+89T2 |
| 97 | 1+9T+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
−8.018940152901382145738569340880, −7.74899505596805586378443603872, −7.05112330546481534770556673989, −5.70313849210100602167971837618, −5.39443459808881737353947382543, −4.45387158450239639287965451044, −3.61668044150343897118175462775, −2.93760772170531985918527862877, −2.02622705927015647123348692929, −0.47808060415157618940497261117,
0.47808060415157618940497261117, 2.02622705927015647123348692929, 2.93760772170531985918527862877, 3.61668044150343897118175462775, 4.45387158450239639287965451044, 5.39443459808881737353947382543, 5.70313849210100602167971837618, 7.05112330546481534770556673989, 7.74899505596805586378443603872, 8.018940152901382145738569340880