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
\(L(1)\) |
\(\approx\) |
\(1.332196649\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.332196649\) |
\(L(\frac{3}{2})\) |
|
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
−8.837330981889445209170182039093, −8.148757906787414549340162106202, −7.47087342767517045035191839317, −6.66291991597438382728727515181, −5.84989202039108958086386331956, −5.12623689594475374157193482857, −4.00869119702315993882191034148, −2.87212671924328834035750560646, −2.05677714430611077712332199523, −0.805037431148696558011542220461,
0.805037431148696558011542220461, 2.05677714430611077712332199523, 2.87212671924328834035750560646, 4.00869119702315993882191034148, 5.12623689594475374157193482857, 5.84989202039108958086386331956, 6.66291991597438382728727515181, 7.47087342767517045035191839317, 8.148757906787414549340162106202, 8.837330981889445209170182039093