L(s) = 1 | − 1.41i·5-s + i·7-s − 1.41·11-s − 1.41i·17-s − 2i·19-s + 1.41·23-s − 1.00·25-s + 1.41·35-s − 1.41i·41-s − 49-s + 2.00i·55-s − 1.41·71-s − 1.41i·77-s − 2.00·85-s + 1.41i·89-s + ⋯ |
L(s) = 1 | − 1.41i·5-s + i·7-s − 1.41·11-s − 1.41i·17-s − 2i·19-s + 1.41·23-s − 1.00·25-s + 1.41·35-s − 1.41i·41-s − 49-s + 2.00i·55-s − 1.41·71-s − 1.41i·77-s − 2.00·85-s + 1.41i·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9525154182\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9525154182\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + 1.41iT - T^{2} \) |
| 11 | \( 1 + 1.41T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 + 2iT - T^{2} \) |
| 23 | \( 1 - 1.41T + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + 1.41iT - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + 1.41T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.41iT - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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
−8.976154012205755560166354338268, −8.680170265363106022567026767162, −7.64152648503015109272804164024, −6.91614135001548020374024290536, −5.61589766192628420569618602160, −5.01696770359342235398471554338, −4.72460924295937248678957118743, −3.01658380050073067093677843413, −2.32073328036948585527311670018, −0.68315849343281942353411925020,
1.66575440404263102548913708498, 2.95097373307377632743712870248, 3.55734639438890375042461572094, 4.57565632907693409972007276983, 5.75967228228394173153998636269, 6.39954639714155647403833438354, 7.32714659515117966369010023572, 7.77572705966048293672151531354, 8.545693792569315061123293041102, 10.07116775517939938096824052210