| L(s) = 1 | + i·3-s + 2.82i·5-s + 4.24·7-s − 9-s + 4i·11-s − 4.24i·13-s − 2.82·15-s + 6·17-s − 2i·19-s + 4.24i·21-s + 2.82·23-s − 3.00·25-s − i·27-s + 5.65i·29-s + 4.24·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.26i·5-s + 1.60·7-s − 0.333·9-s + 1.20i·11-s − 1.17i·13-s − 0.730·15-s + 1.45·17-s − 0.458i·19-s + 0.925i·21-s + 0.589·23-s − 0.600·25-s − 0.192i·27-s + 1.05i·29-s + 0.762·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.486711456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.486711456\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| good | 5 | \( 1 - 2.82iT - 5T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + 4.24iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 5.65iT - 29T^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 - 4.24iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 5.65iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 4.24iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 + 16T + 73T^{2} \) |
| 79 | \( 1 - 4.24T + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−8.871468987495739236000785490938, −7.951061509397089674860091872809, −7.52235835371112572558523972141, −6.80290934482957724359509431976, −5.61858369500923972448414862179, −5.07749275278430744382109910806, −4.28344279901483659029560629241, −3.20493058653168783849721467678, −2.52357701274805309104199087478, −1.27433698457878474941692804917,
0.974685614633874618053590277976, 1.41316059614432738543676892448, 2.62664407358677306910170320235, 4.01433663479847931119866484569, 4.65657712221329644294247686672, 5.53959668612364512700212826385, 5.98652717458654196769014356936, 7.24555442320685152739322760598, 8.014210696909823863143510177232, 8.337540284549585661488405091504
