L(s) = 1 | + 3·2-s + 4-s + 5·5-s + 7·7-s − 21·8-s + 15·10-s + 60·11-s + 38·13-s + 21·14-s − 71·16-s − 84·17-s + 110·19-s + 5·20-s + 180·22-s + 120·23-s + 25·25-s + 114·26-s + 7·28-s + 162·29-s + 236·31-s − 45·32-s − 252·34-s + 35·35-s − 376·37-s + 330·38-s − 105·40-s − 126·41-s + ⋯ |
L(s) = 1 | + 1.06·2-s + 1/8·4-s + 0.447·5-s + 0.377·7-s − 0.928·8-s + 0.474·10-s + 1.64·11-s + 0.810·13-s + 0.400·14-s − 1.10·16-s − 1.19·17-s + 1.32·19-s + 0.0559·20-s + 1.74·22-s + 1.08·23-s + 1/5·25-s + 0.859·26-s + 0.0472·28-s + 1.03·29-s + 1.36·31-s − 0.248·32-s − 1.27·34-s + 0.169·35-s − 1.67·37-s + 1.40·38-s − 0.415·40-s − 0.479·41-s + ⋯ |
Λ(s)=(=(315s/2ΓC(s)L(s)Λ(4−s)
Λ(s)=(=(315s/2ΓC(s+3/2)L(s)Λ(1−s)
Particular Values
L(2) |
≈ |
3.520292536 |
L(21) |
≈ |
3.520292536 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 5 | 1−pT |
| 7 | 1−pT |
good | 2 | 1−3T+p3T2 |
| 11 | 1−60T+p3T2 |
| 13 | 1−38T+p3T2 |
| 17 | 1+84T+p3T2 |
| 19 | 1−110T+p3T2 |
| 23 | 1−120T+p3T2 |
| 29 | 1−162T+p3T2 |
| 31 | 1−236T+p3T2 |
| 37 | 1+376T+p3T2 |
| 41 | 1+126T+p3T2 |
| 43 | 1+34T+p3T2 |
| 47 | 1+6T+p3T2 |
| 53 | 1−582T+p3T2 |
| 59 | 1−492T+p3T2 |
| 61 | 1+880T+p3T2 |
| 67 | 1+826T+p3T2 |
| 71 | 1+666T+p3T2 |
| 73 | 1+826T+p3T2 |
| 79 | 1+592T+p3T2 |
| 83 | 1−792T+p3T2 |
| 89 | 1−1002T+p3T2 |
| 97 | 1−1442T+p3T2 |
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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
−11.70539205514704003597646040648, −10.43193107180795739663972008163, −9.106395941494474397243163013178, −8.720191915741096461651970571706, −6.91786986868751059335531761891, −6.20093816968590882556250996210, −5.06532400364823597101877458716, −4.15366849174035422445509180431, −3.02796480050487528963941092002, −1.26757353380453565560981955821,
1.26757353380453565560981955821, 3.02796480050487528963941092002, 4.15366849174035422445509180431, 5.06532400364823597101877458716, 6.20093816968590882556250996210, 6.91786986868751059335531761891, 8.720191915741096461651970571706, 9.106395941494474397243163013178, 10.43193107180795739663972008163, 11.70539205514704003597646040648