L(s) = 1 | + 3-s + 5-s + 4.68·7-s + 9-s + 2.29·11-s − 4.97·13-s + 15-s − 2.97·17-s − 2.68·19-s + 4.68·21-s + 2.68·23-s + 25-s + 27-s + 2·29-s + 6.97·31-s + 2.29·33-s + 4.68·35-s − 4.39·37-s − 4.97·39-s + 11.3·41-s + 9.37·43-s + 45-s − 7.27·47-s + 14.9·49-s − 2.97·51-s + 2·53-s + 2.29·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.77·7-s + 0.333·9-s + 0.691·11-s − 1.38·13-s + 0.258·15-s − 0.722·17-s − 0.616·19-s + 1.02·21-s + 0.560·23-s + 0.200·25-s + 0.192·27-s + 0.371·29-s + 1.25·31-s + 0.399·33-s + 0.792·35-s − 0.722·37-s − 0.797·39-s + 1.77·41-s + 1.42·43-s + 0.149·45-s − 1.06·47-s + 2.13·49-s − 0.417·51-s + 0.274·53-s + 0.309·55-s + ⋯ |
Λ(s)=(=(3840s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(3840s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
3.277021879 |
L(21) |
≈ |
3.277021879 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1−T |
| 5 | 1−T |
good | 7 | 1−4.68T+7T2 |
| 11 | 1−2.29T+11T2 |
| 13 | 1+4.97T+13T2 |
| 17 | 1+2.97T+17T2 |
| 19 | 1+2.68T+19T2 |
| 23 | 1−2.68T+23T2 |
| 29 | 1−2T+29T2 |
| 31 | 1−6.97T+31T2 |
| 37 | 1+4.39T+37T2 |
| 41 | 1−11.3T+41T2 |
| 43 | 1−9.37T+43T2 |
| 47 | 1+7.27T+47T2 |
| 53 | 1−2T+53T2 |
| 59 | 1−1.70T+59T2 |
| 61 | 1−4.58T+61T2 |
| 67 | 1+4T+67T2 |
| 71 | 1−0.585T+71T2 |
| 73 | 1−6T+73T2 |
| 79 | 1+1.02T+79T2 |
| 83 | 1−13.3T+83T2 |
| 89 | 1+3.37T+89T2 |
| 97 | 1+3.95T+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
−8.481182275456020529051879428097, −7.83834328087237765869672665770, −7.14926576789022755117576503508, −6.38042052374293757128324753759, −5.28770900701820343220553130160, −4.64250208422423109365138752872, −4.11262789093258421338317985499, −2.64401683932788002558649523917, −2.11046612306839665115103320443, −1.09748708760152999777154399496,
1.09748708760152999777154399496, 2.11046612306839665115103320443, 2.64401683932788002558649523917, 4.11262789093258421338317985499, 4.64250208422423109365138752872, 5.28770900701820343220553130160, 6.38042052374293757128324753759, 7.14926576789022755117576503508, 7.83834328087237765869672665770, 8.481182275456020529051879428097