| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 11-s − 12-s + 13-s − 15-s + 16-s + 6·17-s + 18-s + 4·19-s + 20-s − 22-s − 6·23-s − 24-s − 4·25-s + 26-s − 27-s + 3·29-s − 30-s + 11·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.213·22-s − 1.25·23-s − 0.204·24-s − 4/5·25-s + 0.196·26-s − 0.192·27-s + 0.557·29-s − 0.182·30-s + 1.97·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.898650850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.898650850\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.079464557742516455257811209921, −7.88726865992802290068560784929, −6.67977304881098183980770883507, −6.20968245118849588213967385130, −5.41538490462143317344710809691, −4.94809026004257678850587851082, −3.88685634038772924603182367397, −3.12860410237719280517625053481, −2.03413591199556524513892954961, −0.950220163938610621111634211423,
0.950220163938610621111634211423, 2.03413591199556524513892954961, 3.12860410237719280517625053481, 3.88685634038772924603182367397, 4.94809026004257678850587851082, 5.41538490462143317344710809691, 6.20968245118849588213967385130, 6.67977304881098183980770883507, 7.88726865992802290068560784929, 8.079464557742516455257811209921
