L(s) = 1 | + 0.221·2-s − 7.95·4-s + 15.1·5-s + 7·7-s − 3.53·8-s + 3.36·10-s + 11·11-s + 37.8·13-s + 1.55·14-s + 62.8·16-s − 61.7·17-s + 54.5·19-s − 120.·20-s + 2.44·22-s + 24.4·23-s + 104.·25-s + 8.38·26-s − 55.6·28-s − 16.2·29-s − 190.·31-s + 42.2·32-s − 13.7·34-s + 106.·35-s + 170.·37-s + 12.1·38-s − 53.6·40-s + 78.0·41-s + ⋯ |
L(s) = 1 | + 0.0784·2-s − 0.993·4-s + 1.35·5-s + 0.377·7-s − 0.156·8-s + 0.106·10-s + 0.301·11-s + 0.806·13-s + 0.0296·14-s + 0.981·16-s − 0.881·17-s + 0.658·19-s − 1.34·20-s + 0.0236·22-s + 0.222·23-s + 0.838·25-s + 0.0632·26-s − 0.375·28-s − 0.104·29-s − 1.10·31-s + 0.233·32-s − 0.0691·34-s + 0.512·35-s + 0.758·37-s + 0.0516·38-s − 0.212·40-s + 0.297·41-s + ⋯ |
Λ(s)=(=(693s/2ΓC(s)L(s)Λ(4−s)
Λ(s)=(=(693s/2ΓC(s+3/2)L(s)Λ(1−s)
Particular Values
L(2) |
≈ |
2.370911437 |
L(21) |
≈ |
2.370911437 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1−7T |
| 11 | 1−11T |
good | 2 | 1−0.221T+8T2 |
| 5 | 1−15.1T+125T2 |
| 13 | 1−37.8T+2.19e3T2 |
| 17 | 1+61.7T+4.91e3T2 |
| 19 | 1−54.5T+6.85e3T2 |
| 23 | 1−24.4T+1.21e4T2 |
| 29 | 1+16.2T+2.43e4T2 |
| 31 | 1+190.T+2.97e4T2 |
| 37 | 1−170.T+5.06e4T2 |
| 41 | 1−78.0T+6.89e4T2 |
| 43 | 1−45.5T+7.95e4T2 |
| 47 | 1−273.T+1.03e5T2 |
| 53 | 1+163.T+1.48e5T2 |
| 59 | 1−650.T+2.05e5T2 |
| 61 | 1−257.T+2.26e5T2 |
| 67 | 1+399.T+3.00e5T2 |
| 71 | 1−198.T+3.57e5T2 |
| 73 | 1+226.T+3.89e5T2 |
| 79 | 1−138.T+4.93e5T2 |
| 83 | 1−490.T+5.71e5T2 |
| 89 | 1−221.T+7.04e5T2 |
| 97 | 1−1.59e3T+9.12e5T2 |
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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
−9.915710833910314533740264491382, −9.147149516107749674246821442709, −8.692798754201572366621759761334, −7.48141151762571055623308900467, −6.23669108409491643484820616078, −5.57083378709050319940201679872, −4.66303120868514681315042365420, −3.57392835650029745476681056917, −2.10883319986618011783232019110, −0.935061482595919914194394171323,
0.935061482595919914194394171323, 2.10883319986618011783232019110, 3.57392835650029745476681056917, 4.66303120868514681315042365420, 5.57083378709050319940201679872, 6.23669108409491643484820616078, 7.48141151762571055623308900467, 8.692798754201572366621759761334, 9.147149516107749674246821442709, 9.915710833910314533740264491382