L(s) = 1 | − 1.61i·2-s − 1.61·4-s + 0.618i·7-s + i·8-s − 9-s + 1.61i·13-s + 1.00·14-s + 0.618i·17-s + 1.61i·18-s + 1.61·19-s + 1.61i·23-s + 2.61·26-s − 1.00i·28-s − 0.618·29-s + i·32-s + ⋯ |
L(s) = 1 | − 1.61i·2-s − 1.61·4-s + 0.618i·7-s + i·8-s − 9-s + 1.61i·13-s + 1.00·14-s + 0.618i·17-s + 1.61i·18-s + 1.61·19-s + 1.61i·23-s + 2.61·26-s − 1.00i·28-s − 0.618·29-s + i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8610835248\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8610835248\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 103 | \( 1 + iT \) |
good | 2 | \( 1 + 1.61iT - T^{2} \) |
| 3 | \( 1 + T^{2} \) |
| 7 | \( 1 - 0.618iT - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.61iT - T^{2} \) |
| 17 | \( 1 - 0.618iT - T^{2} \) |
| 19 | \( 1 - 1.61T + T^{2} \) |
| 23 | \( 1 - 1.61iT - T^{2} \) |
| 29 | \( 1 + 0.618T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + 1.61T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + 0.618T + T^{2} \) |
| 61 | \( 1 - 0.618T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - 1.61T + T^{2} \) |
| 83 | \( 1 + 0.618iT - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - 0.618iT - T^{2} \) |
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
−9.253223074129126277524896025731, −8.710418809021807937058789306274, −7.69052579394784197687913513002, −6.66883669401795317600334993741, −5.62985045181703279365311278165, −4.96508385090288477066359231000, −3.79564373334265621820309216227, −3.26019411637008586421651795890, −2.22643944098281531365111010865, −1.45962501456639549771994323563,
0.58474809483828700665199015804, 2.70926795065180795614901110791, 3.60828616715751469062424193785, 4.89599302531962308949200619366, 5.34471136082406711280755671525, 6.06569836045488985398165529213, 6.91300816232583591562772877889, 7.58162541323742514453545539189, 8.155053898861688560750372342533, 8.769091110625717844128368260213