L(s) = 1 | − 8·2-s − 24.3·3-s + 64·4-s − 116.·5-s + 194.·6-s + 1.26e3·7-s − 512·8-s − 1.59e3·9-s + 933.·10-s + 3.68e3·11-s − 1.55e3·12-s − 1.45e4·13-s − 1.01e4·14-s + 2.84e3·15-s + 4.09e3·16-s − 2.39e4·17-s + 1.27e4·18-s − 1.48e3·19-s − 7.46e3·20-s − 3.08e4·21-s − 2.94e4·22-s + 3.46e4·23-s + 1.24e4·24-s − 6.45e4·25-s + 1.16e5·26-s + 9.21e4·27-s + 8.10e4·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.520·3-s + 0.5·4-s − 0.417·5-s + 0.368·6-s + 1.39·7-s − 0.353·8-s − 0.728·9-s + 0.295·10-s + 0.834·11-s − 0.260·12-s − 1.84·13-s − 0.987·14-s + 0.217·15-s + 0.250·16-s − 1.18·17-s + 0.515·18-s − 0.0495·19-s − 0.208·20-s − 0.727·21-s − 0.590·22-s + 0.593·23-s + 0.184·24-s − 0.825·25-s + 1.30·26-s + 0.900·27-s + 0.698·28-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) |
≈ |
0.6565593186 |
L(21) |
≈ |
0.6565593186 |
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+24.3T+2.18e3T2 |
| 5 | 1+116.T+7.81e4T2 |
| 7 | 1−1.26e3T+8.23e5T2 |
| 11 | 1−3.68e3T+1.94e7T2 |
| 13 | 1+1.45e4T+6.27e7T2 |
| 17 | 1+2.39e4T+4.10e8T2 |
| 19 | 1+1.48e3T+8.93e8T2 |
| 23 | 1−3.46e4T+3.40e9T2 |
| 29 | 1+4.79e4T+1.72e10T2 |
| 31 | 1+2.58e5T+2.75e10T2 |
| 37 | 1−3.73e5T+9.49e10T2 |
| 41 | 1−3.29e5T+1.94e11T2 |
| 43 | 1−5.36e5T+2.71e11T2 |
| 47 | 1−6.23e4T+5.06e11T2 |
| 53 | 1+1.06e6T+1.17e12T2 |
| 59 | 1+2.35e6T+2.48e12T2 |
| 61 | 1−3.98e5T+3.14e12T2 |
| 67 | 1+3.21e6T+6.06e12T2 |
| 71 | 1−3.83e6T+9.09e12T2 |
| 73 | 1+4.28e6T+1.10e13T2 |
| 79 | 1−5.12e6T+1.92e13T2 |
| 83 | 1+2.90e6T+2.71e13T2 |
| 89 | 1+1.03e7T+4.42e13T2 |
| 97 | 1−1.17e7T+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
−9.493238120886446332516831831805, −8.888306376044114532218024080843, −7.80920513926631521106519575728, −7.28666583864275565658552493812, −6.10483342011451388547862282900, −5.05707603792514387455766598271, −4.24398143543010063135230630815, −2.61318125640387370724291688577, −1.68147615998146276550705001117, −0.39645870772114069840500024429,
0.39645870772114069840500024429, 1.68147615998146276550705001117, 2.61318125640387370724291688577, 4.24398143543010063135230630815, 5.05707603792514387455766598271, 6.10483342011451388547862282900, 7.28666583864275565658552493812, 7.80920513926631521106519575728, 8.888306376044114532218024080843, 9.493238120886446332516831831805