| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.158347904\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.158347904\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + p T \) |
| good | 2 | \( 1 + p T + p^{3} T^{2} \) |
| 7 | \( 1 - 9 T + p^{3} T^{2} \) |
| 11 | \( 1 + 62 T + p^{3} T^{2} \) |
| 13 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 76 T + p^{3} T^{2} \) |
| 23 | \( 1 - 42 T + p^{3} T^{2} \) |
| 29 | \( 1 + 259 T + p^{3} T^{2} \) |
| 31 | \( 1 + 120 T + p^{3} T^{2} \) |
| 37 | \( 1 - 230 T + p^{3} T^{2} \) |
| 41 | \( 1 - 455 T + p^{3} T^{2} \) |
| 43 | \( 1 - 340 T + p^{3} T^{2} \) |
| 47 | \( 1 + 224 T + p^{3} T^{2} \) |
| 53 | \( 1 - 61 T + p^{3} T^{2} \) |
| 59 | \( 1 + 119 T + p^{3} T^{2} \) |
| 61 | \( 1 + 113 T + p^{3} T^{2} \) |
| 67 | \( 1 + 468 T + p^{3} T^{2} \) |
| 71 | \( 1 - 995 T + p^{3} T^{2} \) |
| 73 | \( 1 - 271 T + p^{3} T^{2} \) |
| 79 | \( 1 - 318 T + p^{3} T^{2} \) |
| 83 | \( 1 - 336 T + p^{3} T^{2} \) |
| 89 | \( 1 + 945 T + p^{3} T^{2} \) |
| 97 | \( 1 - 872 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
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
