L(s) = 1 | + 3-s + 3·5-s − 7-s + 9-s + 11-s + 3·13-s + 3·15-s + 17-s + 6·19-s − 21-s + 2·23-s + 4·25-s + 27-s + 6·29-s + 33-s − 3·35-s + 3·37-s + 3·39-s − 11·43-s + 3·45-s + 49-s + 51-s − 9·53-s + 3·55-s + 6·57-s + 4·59-s − 14·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.34·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.832·13-s + 0.774·15-s + 0.242·17-s + 1.37·19-s − 0.218·21-s + 0.417·23-s + 4/5·25-s + 0.192·27-s + 1.11·29-s + 0.174·33-s − 0.507·35-s + 0.493·37-s + 0.480·39-s − 1.67·43-s + 0.447·45-s + 1/7·49-s + 0.140·51-s − 1.23·53-s + 0.404·55-s + 0.794·57-s + 0.520·59-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.691349256\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.691349256\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 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
−8.249450843914495415196766230174, −7.38974374710019368951637251511, −6.58824481671879943163823271409, −6.07270243779146053508314007946, −5.32387297132512078952664699228, −4.51885448864749248695181997098, −3.36376362546877746648880425653, −2.92629751772699708067181008060, −1.80970280058584031948162357044, −1.09242210930887368002922405627,
1.09242210930887368002922405627, 1.80970280058584031948162357044, 2.92629751772699708067181008060, 3.36376362546877746648880425653, 4.51885448864749248695181997098, 5.32387297132512078952664699228, 6.07270243779146053508314007946, 6.58824481671879943163823271409, 7.38974374710019368951637251511, 8.249450843914495415196766230174