L(s) = 1 | + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)11-s − 13-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s − 29-s + (−0.5 + 0.866i)31-s + (−0.5 − 0.866i)37-s + 41-s − 43-s + (0.5 + 0.866i)47-s + (0.5 − 0.866i)53-s + 55-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)11-s − 13-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s − 29-s + (−0.5 + 0.866i)31-s + (−0.5 − 0.866i)37-s + 41-s − 43-s + (0.5 + 0.866i)47-s + (0.5 − 0.866i)53-s + 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.006754813381 - 0.1064417541i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.006754813381 - 0.1064417541i\) |
\(L(1)\) |
\(\approx\) |
\(0.6800521676 - 0.08666147500i\) |
\(L(1)\) |
\(\approx\) |
\(0.6800521676 - 0.08666147500i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.5 + 0.866i)T \) |
| 53 | \( 1 + (0.5 - 0.866i)T \) |
| 59 | \( 1 + (0.5 - 0.866i)T \) |
| 61 | \( 1 + (0.5 + 0.866i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (0.5 - 0.866i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 - T \) |
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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
−31.368674511005381227840381420673, −29.80766252183609808358544132990, −29.34777381821147052864462108326, −27.69773568100531439319707441164, −26.8644839995237322297507010017, −26.00743259798001839000742870384, −24.64021827284966783671935212404, −23.60447507797590863350105021056, −22.49470919995170236391305571967, −21.62518396146473215930300063324, −20.2128353107232475339331239671, −19.06418132110776230625937436484, −18.28238996385279043486707097944, −16.84995099555567988036650204426, −15.636932014297276923314171481569, −14.61502516102634747192292200952, −13.468581525233280518369833769850, −11.93862849492322997468076996313, −10.95208971904920178626871227717, −9.73173819572569088801931885433, −8.09189499551442968009325324501, −7.042766592911424033333627921150, −5.566062373968980361548994344431, −3.837389875079354389370214599208, −2.46845435737195258379327043307,
0.04679959096731799967284018353, 2.125351501808838728582535582363, 4.150675981361680424033891692212, 5.20410232001639393310358722381, 7.02048814094043550062814007489, 8.24501634381663701308683613511, 9.45249196327773318171175531821, 10.845546833591945450101878187493, 12.33242283594329911685870217437, 12.96616556643733607538851601057, 14.68632243733093243005915353251, 15.67150163533345167330992235800, 16.87114664580238510006331928516, 17.852319489178950777930957223455, 19.419762110180769379467468843422, 20.14869875112966893968664001674, 21.305418146443683504065837110708, 22.54765733549928465050606102517, 23.7779853120497779139569344696, 24.46922636919233841358942839252, 25.78823321894642643506319547038, 26.86331676015131759196298587798, 28.099491438132979578758630321215, 28.65921755634291119545043499734, 30.09296175832345639199155775443