L(s) = 1 | + i·7-s + i·11-s − 13-s − i·17-s − i·19-s + i·23-s − i·29-s − 31-s − 37-s + 41-s − 43-s + i·47-s − 49-s − 53-s − i·59-s + ⋯ |
L(s) = 1 | + i·7-s + i·11-s − 13-s − i·17-s − i·19-s + i·23-s − i·29-s − 31-s − 37-s + 41-s − 43-s + i·47-s − 49-s − 53-s − i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01774295833 + 0.2201043593i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01774295833 + 0.2201043593i\) |
\(L(1)\) |
\(\approx\) |
\(0.8006816524 + 0.1299327450i\) |
\(L(1)\) |
\(\approx\) |
\(0.8006816524 + 0.1299327450i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 \) |
| 19 | \( 1 + iT \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
| 31 | \( 1 \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 \) |
| 53 | \( 1 \) |
| 59 | \( 1 \) |
| 61 | \( 1 - iT \) |
| 67 | \( 1 \) |
| 71 | \( 1 - iT \) |
| 73 | \( 1 \) |
| 79 | \( 1 \) |
| 83 | \( 1 \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 \) |
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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
−25.53013506943827650354809614381, −24.35448956351720493959416707317, −23.799124490345096442243543840066, −22.69415485964946117528816395027, −21.8062794558502212993060215966, −20.82812575038601963451749580776, −19.84002141146182686254803893507, −19.09174756785500680579509143141, −17.95635781104975707976863681333, −16.776876929413941049784270925797, −16.42289947786992869914647172634, −14.84019857876283810923944235351, −14.173749301747250442467903146405, −13.07725030293458676183133630318, −12.12903911854059134985835873862, −10.79167885769729210289615557803, −10.21746276269951579529996333390, −8.82145582895739037561101673732, −7.7975221219391697813086167184, −6.750916539888534341115480506075, −5.56490631559548881805191998779, −4.24838288224369724719901116392, −3.2131894540678199562896326534, −1.56639126664496161295244757372, −0.067448963105514796776224935792,
1.93280413186258102059531705215, 2.93901747416417136651954290084, 4.621676038075325180016319171248, 5.44617472604229192354173661475, 6.856356250783098517912359124230, 7.76596298664072311559810045205, 9.21761117713792563211887966518, 9.72518156383206779424800679814, 11.264775504178352776325736930257, 12.10208499329078841158158557880, 12.97627782640295995054751797394, 14.24392359834627117801354145095, 15.22066547681737834824528860200, 15.87409173241923396878953210208, 17.29179090092328734275342797482, 17.93866639241588472281856820022, 19.03575926702200180079498079624, 19.90000965085716447624355569896, 20.92784548357680384550265895239, 21.945228524290550003250763136974, 22.59364366913973131402460651557, 23.73603470207056493742201265511, 24.75438136639864701880813427916, 25.38686756170531515177257897289, 26.39468146225972509785714739611