L(s) = 1 | − 27·3-s + 100·5-s − 343·7-s + 729·9-s + 2.77e3·11-s − 3.29e3·13-s − 2.70e3·15-s + 5.90e3·17-s + 6.64e3·19-s + 9.26e3·21-s + 1.98e3·23-s − 6.81e4·25-s − 1.96e4·27-s − 2.08e5·29-s − 1.17e5·31-s − 7.48e4·33-s − 3.43e4·35-s − 3.35e5·37-s + 8.89e4·39-s − 2.65e5·41-s − 9.32e4·43-s + 7.29e4·45-s − 6.57e5·47-s + 1.17e5·49-s − 1.59e5·51-s − 6.08e5·53-s + 2.77e5·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.357·5-s − 0.377·7-s + 1/3·9-s + 0.628·11-s − 0.415·13-s − 0.206·15-s + 0.291·17-s + 0.222·19-s + 0.218·21-s + 0.0339·23-s − 0.871·25-s − 0.192·27-s − 1.58·29-s − 0.710·31-s − 0.362·33-s − 0.135·35-s − 1.08·37-s + 0.240·39-s − 0.601·41-s − 0.178·43-s + 0.119·45-s − 0.923·47-s + 1/7·49-s − 0.168·51-s − 0.561·53-s + 0.224·55-s + ⋯ |
Λ(s)=(=(84s/2ΓC(s)L(s)−Λ(8−s)
Λ(s)=(=(84s/2ΓC(s+7/2)L(s)−Λ(1−s)
Particular Values
L(4) |
= |
0 |
L(21) |
= |
0 |
L(29) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+p3T |
| 7 | 1+p3T |
good | 5 | 1−4p2T+p7T2 |
| 11 | 1−2774T+p7T2 |
| 13 | 1+3294T+p7T2 |
| 17 | 1−5900T+p7T2 |
| 19 | 1−6644T+p7T2 |
| 23 | 1−1982T+p7T2 |
| 29 | 1+208106T+p7T2 |
| 31 | 1+117792T+p7T2 |
| 37 | 1+335686T+p7T2 |
| 41 | 1+265488T+p7T2 |
| 43 | 1+93292T+p7T2 |
| 47 | 1+657516T+p7T2 |
| 53 | 1+608718T+p7T2 |
| 59 | 1+536120T+p7T2 |
| 61 | 1+1797090T+p7T2 |
| 67 | 1−2123176T+p7T2 |
| 71 | 1+1191214T+p7T2 |
| 73 | 1−1056430T+p7T2 |
| 79 | 1−998484T+p7T2 |
| 83 | 1−3898004T+p7T2 |
| 89 | 1+4622352T+p7T2 |
| 97 | 1−15287710T+p7T2 |
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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
−12.26327752688485113692119052833, −11.28963543910156009299305564943, −10.05912654876854082151654391176, −9.168632393704708287271195018394, −7.51876199311847863803506807154, −6.33469311027517854523544770523, −5.21606738572037131193301105895, −3.62403721098224109192558892109, −1.72781302756464489008368950815, 0,
1.72781302756464489008368950815, 3.62403721098224109192558892109, 5.21606738572037131193301105895, 6.33469311027517854523544770523, 7.51876199311847863803506807154, 9.168632393704708287271195018394, 10.05912654876854082151654391176, 11.28963543910156009299305564943, 12.26327752688485113692119052833