L(s) = 1 | + (0.0747 − 0.997i)2-s + (−0.988 − 0.149i)4-s + (0.826 + 0.563i)5-s + (0.988 − 0.149i)7-s + (−0.222 + 0.974i)8-s + (0.623 − 0.781i)10-s + (−0.955 − 0.294i)11-s + (−0.733 + 0.680i)13-s + (−0.0747 − 0.997i)14-s + (0.955 + 0.294i)16-s − 17-s + (0.623 − 0.781i)19-s + (−0.733 − 0.680i)20-s + (−0.365 + 0.930i)22-s + (0.365 + 0.930i)25-s + (0.623 + 0.781i)26-s + ⋯ |
L(s) = 1 | + (0.0747 − 0.997i)2-s + (−0.988 − 0.149i)4-s + (0.826 + 0.563i)5-s + (0.988 − 0.149i)7-s + (−0.222 + 0.974i)8-s + (0.623 − 0.781i)10-s + (−0.955 − 0.294i)11-s + (−0.733 + 0.680i)13-s + (−0.0747 − 0.997i)14-s + (0.955 + 0.294i)16-s − 17-s + (0.623 − 0.781i)19-s + (−0.733 − 0.680i)20-s + (−0.365 + 0.930i)22-s + (0.365 + 0.930i)25-s + (0.623 + 0.781i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.394530063 + 0.2696905915i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.394530063 + 0.2696905915i\) |
\(L(1)\) |
\(\approx\) |
\(1.021195421 - 0.3232514368i\) |
\(L(1)\) |
\(\approx\) |
\(1.021195421 - 0.3232514368i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (0.0747 - 0.997i)T \) |
| 5 | \( 1 + (0.826 + 0.563i)T \) |
| 7 | \( 1 + (0.988 - 0.149i)T \) |
| 11 | \( 1 + (-0.955 - 0.294i)T \) |
| 13 | \( 1 + (-0.733 + 0.680i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + (0.623 - 0.781i)T \) |
| 31 | \( 1 + (-0.826 - 0.563i)T \) |
| 37 | \( 1 + (-0.222 + 0.974i)T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.826 - 0.563i)T \) |
| 47 | \( 1 + (0.955 + 0.294i)T \) |
| 53 | \( 1 + (-0.900 + 0.433i)T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.988 + 0.149i)T \) |
| 67 | \( 1 + (-0.955 + 0.294i)T \) |
| 71 | \( 1 + (0.222 + 0.974i)T \) |
| 73 | \( 1 + (0.900 + 0.433i)T \) |
| 79 | \( 1 + (-0.733 - 0.680i)T \) |
| 83 | \( 1 + (0.365 + 0.930i)T \) |
| 89 | \( 1 + (0.900 - 0.433i)T \) |
| 97 | \( 1 + (0.365 + 0.930i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.52882375547897814422464841548, −17.19171175794126507400461477622, −16.287305872492072474266674648137, −15.77945160558198264997437473553, −15.0328148280057256522293210936, −14.45094693472030964618437382351, −13.88787091505871058798823134933, −13.15662786557734746414820246310, −12.63258873670637611487650859639, −12.03601972410102622171214513178, −10.82648379057927751114785582297, −10.30847859230711565928662819756, −9.44566916484480331183614211627, −8.98541472894944185389643243655, −8.06514899818669172154552679644, −7.765555506361905997568287284683, −6.95432621099809389357468321724, −5.98782974533667837650415905472, −5.42938109987998282739259137413, −4.92176410954058811309333703558, −4.41832731590685513851072411683, −3.25236790513750226294027541039, −2.26322245552144139329625299518, −1.534589599205617986001804395280, −0.37346515289443799456105475082,
0.9378657391552326313549986211, 1.94512667902377117111467401919, 2.34314393442689524117317980644, 3.005845840768558340122232053456, 4.04579272446075214824462547122, 4.80739131975154495938745878402, 5.28698304989038820731420556314, 6.0501125866580609088708457610, 7.15330198615529972088089738673, 7.64820991583120303850615158820, 8.76176491315663515289701423236, 9.11336959312428472264066291402, 9.98212371426498924737920583785, 10.58804267289111811644163534830, 11.10116598879893165196527912994, 11.62579246122459581896985121682, 12.45074235169579308014235953201, 13.305334799814905121366796526649, 13.73651287192926990978766494464, 14.20419262544790603373689619654, 14.98242044342891991598416362704, 15.56675621654109798261084663311, 16.75867396166318644241688983272, 17.470606261373113367309397603281, 17.72198716107686095177971768812