L(s) = 1 | + 8·2-s + 64.1·3-s + 64·4-s − 288.·5-s + 513.·6-s + 1.13e3·7-s + 512·8-s + 1.92e3·9-s − 2.30e3·10-s + 6.91e3·11-s + 4.10e3·12-s + 8.35e3·13-s + 9.09e3·14-s − 1.84e4·15-s + 4.09e3·16-s − 1.60e4·17-s + 1.54e4·18-s + 1.85e4·19-s − 1.84e4·20-s + 7.29e4·21-s + 5.53e4·22-s + 3.41e4·23-s + 3.28e4·24-s + 4.98e3·25-s + 6.68e4·26-s − 1.67e4·27-s + 7.27e4·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.37·3-s + 0.5·4-s − 1.03·5-s + 0.969·6-s + 1.25·7-s + 0.353·8-s + 0.880·9-s − 0.729·10-s + 1.56·11-s + 0.685·12-s + 1.05·13-s + 0.885·14-s − 1.41·15-s + 0.250·16-s − 0.791·17-s + 0.622·18-s + 0.619·19-s − 0.515·20-s + 1.71·21-s + 1.10·22-s + 0.585·23-s + 0.484·24-s + 0.0637·25-s + 0.745·26-s − 0.163·27-s + 0.626·28-s + ⋯ |
Λ(s)=(=(538s/2ΓC(s)L(s)Λ(8−s)
Λ(s)=(=(538s/2ΓC(s+7/2)L(s)Λ(1−s)
Particular Values
L(4) |
≈ |
7.458632802 |
L(21) |
≈ |
7.458632802 |
L(29) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1−8T |
| 269 | 1−1.94e7T |
good | 3 | 1−64.1T+2.18e3T2 |
| 5 | 1+288.T+7.81e4T2 |
| 7 | 1−1.13e3T+8.23e5T2 |
| 11 | 1−6.91e3T+1.94e7T2 |
| 13 | 1−8.35e3T+6.27e7T2 |
| 17 | 1+1.60e4T+4.10e8T2 |
| 19 | 1−1.85e4T+8.93e8T2 |
| 23 | 1−3.41e4T+3.40e9T2 |
| 29 | 1+3.80e4T+1.72e10T2 |
| 31 | 1+9.23e4T+2.75e10T2 |
| 37 | 1−4.00e5T+9.49e10T2 |
| 41 | 1+1.63e5T+1.94e11T2 |
| 43 | 1−6.23e5T+2.71e11T2 |
| 47 | 1+1.64e5T+5.06e11T2 |
| 53 | 1−8.51e5T+1.17e12T2 |
| 59 | 1+5.90e5T+2.48e12T2 |
| 61 | 1+2.22e6T+3.14e12T2 |
| 67 | 1+3.77e6T+6.06e12T2 |
| 71 | 1+6.37e5T+9.09e12T2 |
| 73 | 1−2.41e6T+1.10e13T2 |
| 79 | 1+2.94e6T+1.92e13T2 |
| 83 | 1−9.83e6T+2.71e13T2 |
| 89 | 1−4.92e6T+4.42e13T2 |
| 97 | 1+1.81e6T+8.07e13T2 |
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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
−9.290604336243382862236249389259, −8.720134752356523361398541740424, −7.88577939933376008150199466666, −7.25485397904000869844688870902, −6.04652151754761743201002704101, −4.56869412541408041506587890174, −3.96009925249341220674984366973, −3.25016370829111510950547266462, −1.96933230367495089676940005686, −1.09254001688510598134945952326,
1.09254001688510598134945952326, 1.96933230367495089676940005686, 3.25016370829111510950547266462, 3.96009925249341220674984366973, 4.56869412541408041506587890174, 6.04652151754761743201002704101, 7.25485397904000869844688870902, 7.88577939933376008150199466666, 8.720134752356523361398541740424, 9.290604336243382862236249389259