L(s) = 1 | − 8·2-s − 19.6·3-s + 64·4-s − 242.·5-s + 157.·6-s − 1.44e3·7-s − 512·8-s − 1.80e3·9-s + 1.93e3·10-s − 2.21e3·11-s − 1.25e3·12-s − 1.40e4·13-s + 1.15e4·14-s + 4.76e3·15-s + 4.09e3·16-s − 1.99e4·17-s + 1.44e4·18-s + 3.07e4·19-s − 1.55e4·20-s + 2.83e4·21-s + 1.77e4·22-s + 6.63e4·23-s + 1.00e4·24-s − 1.94e4·25-s + 1.12e5·26-s + 7.84e4·27-s − 9.22e4·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.420·3-s + 0.5·4-s − 0.866·5-s + 0.297·6-s − 1.58·7-s − 0.353·8-s − 0.823·9-s + 0.612·10-s − 0.502·11-s − 0.210·12-s − 1.77·13-s + 1.12·14-s + 0.364·15-s + 0.250·16-s − 0.986·17-s + 0.582·18-s + 1.02·19-s − 0.433·20-s + 0.667·21-s + 0.355·22-s + 1.13·23-s + 0.148·24-s − 0.248·25-s + 1.25·26-s + 0.766·27-s − 0.794·28-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 |
L(21) |
= |
0 |
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+19.6T+2.18e3T2 |
| 5 | 1+242.T+7.81e4T2 |
| 7 | 1+1.44e3T+8.23e5T2 |
| 11 | 1+2.21e3T+1.94e7T2 |
| 13 | 1+1.40e4T+6.27e7T2 |
| 17 | 1+1.99e4T+4.10e8T2 |
| 19 | 1−3.07e4T+8.93e8T2 |
| 23 | 1−6.63e4T+3.40e9T2 |
| 29 | 1−2.43e5T+1.72e10T2 |
| 31 | 1−5.66e4T+2.75e10T2 |
| 37 | 1+5.31e5T+9.49e10T2 |
| 41 | 1−3.59e5T+1.94e11T2 |
| 43 | 1−5.52e5T+2.71e11T2 |
| 47 | 1+2.99e5T+5.06e11T2 |
| 53 | 1−2.10e5T+1.17e12T2 |
| 59 | 1−2.94e6T+2.48e12T2 |
| 61 | 1+2.89e6T+3.14e12T2 |
| 67 | 1−1.08e6T+6.06e12T2 |
| 71 | 1+4.83e6T+9.09e12T2 |
| 73 | 1+4.65e6T+1.10e13T2 |
| 79 | 1+6.00e5T+1.92e13T2 |
| 83 | 1−4.21e6T+2.71e13T2 |
| 89 | 1+6.58e6T+4.42e13T2 |
| 97 | 1+1.36e7T+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
−9.297188512502135514457579846562, −8.442917379104110740816191548732, −7.30096373562027807376639611486, −6.82760036111758859431930650546, −5.69562166299809962121910497104, −4.62797919893298041782563168210, −3.11195367033617062222828259717, −2.62889722120193749705310329577, −0.61389306659533056323248079613, 0,
0.61389306659533056323248079613, 2.62889722120193749705310329577, 3.11195367033617062222828259717, 4.62797919893298041782563168210, 5.69562166299809962121910497104, 6.82760036111758859431930650546, 7.30096373562027807376639611486, 8.442917379104110740816191548732, 9.297188512502135514457579846562