L(s) = 1 | − 2·2-s + 3·3-s − 4·4-s − 6·6-s + 9·7-s + 24·8-s + 9·9-s − 62·11-s − 12·12-s − 38·13-s − 18·14-s − 16·16-s + 76·17-s − 18·18-s − 19·19-s + 27·21-s + 124·22-s + 42·23-s + 72·24-s + 76·26-s + 27·27-s − 36·28-s − 259·29-s − 120·31-s − 160·32-s − 186·33-s − 152·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.485·7-s + 1.06·8-s + 1/3·9-s − 1.69·11-s − 0.288·12-s − 0.810·13-s − 0.343·14-s − 1/4·16-s + 1.08·17-s − 0.235·18-s − 0.229·19-s + 0.280·21-s + 1.20·22-s + 0.380·23-s + 0.612·24-s + 0.573·26-s + 0.192·27-s − 0.242·28-s − 1.65·29-s − 0.695·31-s − 0.883·32-s − 0.981·33-s − 0.766·34-s + ⋯ |
Λ(s)=(=(1425s/2ΓC(s)L(s)Λ(4−s)
Λ(s)=(=(1425s/2ΓC(s+3/2)L(s)Λ(1−s)
Particular Values
L(2) |
≈ |
1.158347904 |
L(21) |
≈ |
1.158347904 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1−pT |
| 5 | 1 |
| 19 | 1+pT |
good | 2 | 1+pT+p3T2 |
| 7 | 1−9T+p3T2 |
| 11 | 1+62T+p3T2 |
| 13 | 1+38T+p3T2 |
| 17 | 1−76T+p3T2 |
| 23 | 1−42T+p3T2 |
| 29 | 1+259T+p3T2 |
| 31 | 1+120T+p3T2 |
| 37 | 1−230T+p3T2 |
| 41 | 1−455T+p3T2 |
| 43 | 1−340T+p3T2 |
| 47 | 1+224T+p3T2 |
| 53 | 1−61T+p3T2 |
| 59 | 1+119T+p3T2 |
| 61 | 1+113T+p3T2 |
| 67 | 1+468T+p3T2 |
| 71 | 1−995T+p3T2 |
| 73 | 1−271T+p3T2 |
| 79 | 1−318T+p3T2 |
| 83 | 1−336T+p3T2 |
| 89 | 1+945T+p3T2 |
| 97 | 1−872T+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
−9.399945023670959942825222776432, −8.199482367250249069598863324295, −7.76953141331646631384663746627, −7.33068586707932082582129976077, −5.65359574885708013020421081776, −5.01234752363427079419556268095, −4.08163550688887804582919107818, −2.86733010096380091711448167725, −1.88118338601509939901584226944, −0.57324611595455827646089408524,
0.57324611595455827646089408524, 1.88118338601509939901584226944, 2.86733010096380091711448167725, 4.08163550688887804582919107818, 5.01234752363427079419556268095, 5.65359574885708013020421081776, 7.33068586707932082582129976077, 7.76953141331646631384663746627, 8.199482367250249069598863324295, 9.399945023670959942825222776432