L(s) = 1 | + 3-s + 5-s + 2·7-s + 9-s + 2·11-s − 13-s + 15-s − 2·17-s + 2·19-s + 2·21-s + 4·23-s + 25-s + 27-s + 2·29-s + 2·31-s + 2·33-s + 2·35-s − 6·37-s − 39-s − 2·41-s + 8·43-s + 45-s + 6·47-s − 3·49-s − 2·51-s + 6·53-s + 2·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.603·11-s − 0.277·13-s + 0.258·15-s − 0.485·17-s + 0.458·19-s + 0.436·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.359·31-s + 0.348·33-s + 0.338·35-s − 0.986·37-s − 0.160·39-s − 0.312·41-s + 1.21·43-s + 0.149·45-s + 0.875·47-s − 3/7·49-s − 0.280·51-s + 0.824·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.981076957\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.981076957\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.778088763576398463997365708311, −8.018928906875004648120774295537, −7.20458422642905818237419189484, −6.58826857234209685731857182071, −5.55070733514450800796069337739, −4.80615085170454429583454161360, −4.00463706082617694154185810416, −2.96494003213633984496237686200, −2.07219421254708560127484501053, −1.10126722073891486963638052361,
1.10126722073891486963638052361, 2.07219421254708560127484501053, 2.96494003213633984496237686200, 4.00463706082617694154185810416, 4.80615085170454429583454161360, 5.55070733514450800796069337739, 6.58826857234209685731857182071, 7.20458422642905818237419189484, 8.018928906875004648120774295537, 8.778088763576398463997365708311