L(s) = 1 | − 1.61i·2-s + 2.61i·3-s − 0.618·4-s + 4.23·6-s − 3.85i·7-s − 2.23i·8-s − 3.85·9-s + 0.618·11-s − 1.61i·12-s − 5.47i·13-s − 6.23·14-s − 4.85·16-s + 1.47i·17-s + 6.23i·18-s + 0.854·19-s + ⋯ |
L(s) = 1 | − 1.14i·2-s + 1.51i·3-s − 0.309·4-s + 1.72·6-s − 1.45i·7-s − 0.790i·8-s − 1.28·9-s + 0.186·11-s − 0.467i·12-s − 1.51i·13-s − 1.66·14-s − 1.21·16-s + 0.357i·17-s + 1.46i·18-s + 0.195·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.05000 - 1.05000i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05000 - 1.05000i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + 1.61iT - 2T^{2} \) |
| 3 | \( 1 - 2.61iT - 3T^{2} \) |
| 7 | \( 1 + 3.85iT - 7T^{2} \) |
| 11 | \( 1 - 0.618T + 11T^{2} \) |
| 13 | \( 1 + 5.47iT - 13T^{2} \) |
| 17 | \( 1 - 1.47iT - 17T^{2} \) |
| 19 | \( 1 - 0.854T + 19T^{2} \) |
| 23 | \( 1 + 1.85iT - 23T^{2} \) |
| 29 | \( 1 - 2.76T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 + 0.472iT - 43T^{2} \) |
| 47 | \( 1 + 11.6iT - 47T^{2} \) |
| 53 | \( 1 - 8.47iT - 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 + 8.85T + 61T^{2} \) |
| 67 | \( 1 - 11.4iT - 67T^{2} \) |
| 71 | \( 1 + 3.32T + 71T^{2} \) |
| 73 | \( 1 + 0.145iT - 73T^{2} \) |
| 79 | \( 1 - 6.70T + 79T^{2} \) |
| 83 | \( 1 - 12.9iT - 83T^{2} \) |
| 89 | \( 1 + 3.61T + 89T^{2} \) |
| 97 | \( 1 - 10.6iT - 97T^{2} \) |
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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
−10.44241997369762634492752479991, −10.05601237120709182056418423236, −9.114684426929364030822942682213, −7.937229155136574955731386822714, −6.83152524706206485243017744116, −5.45087009410928141423052175192, −4.27261306585304896282519418034, −3.76022017568674038178614848138, −2.80940249888711560477957393448, −0.842135822189655507526748284758,
1.76722983934767260449883778584, 2.66641329302321302120501466047, 4.77772527176458139093983580739, 5.89941183800592003666777930235, 6.41308904945353981472817482865, 7.14770745088067745770262634124, 7.973794096730107489935717538577, 8.741888761674668496697042790001, 9.446137529835488145710606148807, 11.30200565097215419638790205175