L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s + 11-s − 2·13-s + 16-s − 6·17-s − 5·19-s − 20-s + 22-s − 3·23-s − 4·25-s − 2·26-s − 2·29-s + 5·31-s + 32-s − 6·34-s + 3·37-s − 5·38-s − 40-s + 3·41-s − 2·43-s + 44-s − 3·46-s − 10·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s + 0.301·11-s − 0.554·13-s + 1/4·16-s − 1.45·17-s − 1.14·19-s − 0.223·20-s + 0.213·22-s − 0.625·23-s − 4/5·25-s − 0.392·26-s − 0.371·29-s + 0.898·31-s + 0.176·32-s − 1.02·34-s + 0.493·37-s − 0.811·38-s − 0.158·40-s + 0.468·41-s − 0.304·43-s + 0.150·44-s − 0.442·46-s − 1.45·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 17 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.248757321277682355869838998162, −7.78683043510655917697662895053, −6.58542605891929715263111036574, −6.41604904917129958586331233033, −5.19809769267742497006601301083, −4.39150514718666308751838713709, −3.87625921116180662778116695658, −2.68531423090995241408035903546, −1.83563020255482523915816336597, 0,
1.83563020255482523915816336597, 2.68531423090995241408035903546, 3.87625921116180662778116695658, 4.39150514718666308751838713709, 5.19809769267742497006601301083, 6.41604904917129958586331233033, 6.58542605891929715263111036574, 7.78683043510655917697662895053, 8.248757321277682355869838998162