L(s) = 1 | + 2-s + 3-s + 4-s − 3·5-s + 6-s + 7-s + 8-s + 9-s − 3·10-s − 11-s + 12-s + 13-s + 14-s − 3·15-s + 16-s + 18-s + 6·19-s − 3·20-s + 21-s − 22-s + 2·23-s + 24-s + 4·25-s + 26-s + 27-s + 28-s + 2·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s − 0.301·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.235·18-s + 1.37·19-s − 0.670·20-s + 0.218·21-s − 0.213·22-s + 0.417·23-s + 0.204·24-s + 4/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.542811501\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.542811501\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−16.19591199835162, −15.60369683713632, −15.15085526967185, −14.86771110489135, −13.91770227364711, −13.72495734266344, −13.01442875809455, −12.27807199858012, −11.81071187960779, −11.46192488050314, −10.67577454362164, −10.15745468029819, −9.269361166780003, −8.544717800733227, −8.060812719583546, −7.448202440974176, −7.041395795914875, −6.197694310845295, −5.103251033895449, −4.905765004235419, −3.854065870528248, −3.512113468661039, −2.825409206117101, −1.807893469283785, −0.7544348244271245,
0.7544348244271245, 1.807893469283785, 2.825409206117101, 3.512113468661039, 3.854065870528248, 4.905765004235419, 5.103251033895449, 6.197694310845295, 7.041395795914875, 7.448202440974176, 8.060812719583546, 8.544717800733227, 9.269361166780003, 10.15745468029819, 10.67577454362164, 11.46192488050314, 11.81071187960779, 12.27807199858012, 13.01442875809455, 13.72495734266344, 13.91770227364711, 14.86771110489135, 15.15085526967185, 15.60369683713632, 16.19591199835162