L(s) = 1 | + (−0.5 − 0.866i)5-s − 7-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s − 23-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)29-s + (0.5 + 0.866i)31-s + (0.5 + 0.866i)35-s + (−0.5 + 0.866i)37-s + 41-s − 43-s + (0.5 − 0.866i)47-s + 49-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)5-s − 7-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s − 23-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)29-s + (0.5 + 0.866i)31-s + (0.5 + 0.866i)35-s + (−0.5 + 0.866i)37-s + 41-s − 43-s + (0.5 − 0.866i)47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.270611090 - 0.05139539412i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.270611090 - 0.05139539412i\) |
\(L(1)\) |
\(\approx\) |
\(0.8684744642 - 0.07037359885i\) |
\(L(1)\) |
\(\approx\) |
\(0.8684744642 - 0.07037359885i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 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 | \( 1 + T \) |
| 59 | \( 1 + (0.5 - 0.866i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.5 - 0.866i)T \) |
| 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
−23.74398337286135644741638504222, −22.45861570465071904573204948511, −22.3078185174501549883698532949, −21.330372335704777329631036184112, −19.80710779320238391149655294421, −19.569957614287508992180392462292, −18.640089556839905429149957572467, −17.78450984652532931921388557727, −16.63626372073288745341385265648, −15.87475152675398531991385782171, −15.09823584238685607819824786768, −14.0954831984658265565526272038, −13.28852423936666995086986601910, −12.22196482696330269917168702160, −11.28524677089255265741108396633, −10.53562792123008656451631502593, −9.49616997481390137206487997492, −8.531617962803110231916813978935, −7.38135145586178667417097809865, −6.51902234826495975810918572194, −5.78035837889664774526000086164, −4.08996443575578192143701065174, −3.40090599895348654843692509379, −2.350125906913682785332306928, −0.56025994344595041070535702711,
0.64540909289734306680930853941, 2.02633419989479749594060503007, 3.47645747867664351202943181300, 4.3259887001731795694132341824, 5.397417677438080574318859562, 6.561031999426042010992485032493, 7.482366004640878160252159757826, 8.54629685347190928559943763642, 9.514711544661627228661548537787, 10.11677962371534377685996452940, 11.713037919637006630721010776109, 12.16082474651815920705847171844, 13.10477507456313627224627837645, 13.95496784543246855415591728661, 15.209022336950574427209665704876, 15.96505018840357782601229573641, 16.6044043282723266933458856975, 17.55624220031618708119975204552, 18.59476422371914580248544596190, 19.5835467562051546036482289134, 20.16450413164897102254252030075, 20.86673106787441800359673096066, 22.17294791418603600433376530497, 22.80573995253330650689105106760, 23.5228044874960109223587523801