L(s) = 1 | + 3·3-s − 3.29·5-s + 25.8·7-s + 9·9-s − 8.83·11-s + 13·13-s − 9.87·15-s − 10.2·17-s − 119.·19-s + 77.6·21-s + 141.·23-s − 114.·25-s + 27·27-s − 170.·29-s − 226.·31-s − 26.5·33-s − 85.1·35-s + 225.·37-s + 39·39-s − 274.·41-s − 111.·43-s − 29.6·45-s − 156.·47-s + 326.·49-s − 30.7·51-s + 85.1·53-s + 29.0·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.294·5-s + 1.39·7-s + 0.333·9-s − 0.242·11-s + 0.277·13-s − 0.169·15-s − 0.146·17-s − 1.44·19-s + 0.806·21-s + 1.28·23-s − 0.913·25-s + 0.192·27-s − 1.09·29-s − 1.31·31-s − 0.139·33-s − 0.411·35-s + 1.00·37-s + 0.160·39-s − 1.04·41-s − 0.394·43-s − 0.0981·45-s − 0.485·47-s + 0.951·49-s − 0.0844·51-s + 0.220·53-s + 0.0712·55-s + ⋯ |
Λ(s)=(=(2496s/2ΓC(s)L(s)−Λ(4−s)
Λ(s)=(=(2496s/2ΓC(s+3/2)L(s)−Λ(1−s)
Particular Values
L(2) |
= |
0 |
L(21) |
= |
0 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1−3T |
| 13 | 1−13T |
good | 5 | 1+3.29T+125T2 |
| 7 | 1−25.8T+343T2 |
| 11 | 1+8.83T+1.33e3T2 |
| 17 | 1+10.2T+4.91e3T2 |
| 19 | 1+119.T+6.85e3T2 |
| 23 | 1−141.T+1.21e4T2 |
| 29 | 1+170.T+2.43e4T2 |
| 31 | 1+226.T+2.97e4T2 |
| 37 | 1−225.T+5.06e4T2 |
| 41 | 1+274.T+6.89e4T2 |
| 43 | 1+111.T+7.95e4T2 |
| 47 | 1+156.T+1.03e5T2 |
| 53 | 1−85.1T+1.48e5T2 |
| 59 | 1+889.T+2.05e5T2 |
| 61 | 1+463.T+2.26e5T2 |
| 67 | 1+459.T+3.00e5T2 |
| 71 | 1−560.T+3.57e5T2 |
| 73 | 1−784.T+3.89e5T2 |
| 79 | 1+241.T+4.93e5T2 |
| 83 | 1+1.27e3T+5.71e5T2 |
| 89 | 1−1.08e3T+7.04e5T2 |
| 97 | 1+79.9T+9.12e5T2 |
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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
−8.121617874753475687427009663906, −7.67266448524076802886933105197, −6.82475704958091864019907538786, −5.77717529354572969361287225591, −4.87192587471788543110771209093, −4.21918831308961056947452332425, −3.31123904195694607949534632809, −2.12979498373865814289494293407, −1.48880635594307386183114939383, 0,
1.48880635594307386183114939383, 2.12979498373865814289494293407, 3.31123904195694607949534632809, 4.21918831308961056947452332425, 4.87192587471788543110771209093, 5.77717529354572969361287225591, 6.82475704958091864019907538786, 7.67266448524076802886933105197, 8.121617874753475687427009663906