L(s) = 1 | + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)11-s + (0.173 − 0.984i)13-s + (−0.939 − 0.342i)17-s + (−0.766 − 0.642i)23-s + (−0.939 + 0.342i)29-s + (0.5 − 0.866i)31-s + 37-s + (−0.173 − 0.984i)41-s + (−0.766 + 0.642i)43-s + (0.939 − 0.342i)47-s + (−0.5 − 0.866i)49-s + (−0.766 − 0.642i)53-s + (0.939 + 0.342i)59-s + (−0.766 − 0.642i)61-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)11-s + (0.173 − 0.984i)13-s + (−0.939 − 0.342i)17-s + (−0.766 − 0.642i)23-s + (−0.939 + 0.342i)29-s + (0.5 − 0.866i)31-s + 37-s + (−0.173 − 0.984i)41-s + (−0.766 + 0.642i)43-s + (0.939 − 0.342i)47-s + (−0.5 − 0.866i)49-s + (−0.766 − 0.642i)53-s + (0.939 + 0.342i)59-s + (−0.766 − 0.642i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9369995324 - 0.5508956363i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9369995324 - 0.5508956363i\) |
\(L(1)\) |
\(\approx\) |
\(0.9323125819 + 0.02353623058i\) |
\(L(1)\) |
\(\approx\) |
\(0.9323125819 + 0.02353623058i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.173 - 0.984i)T \) |
| 17 | \( 1 + (-0.939 - 0.342i)T \) |
| 23 | \( 1 + (-0.766 - 0.642i)T \) |
| 29 | \( 1 + (-0.939 + 0.342i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (-0.173 - 0.984i)T \) |
| 43 | \( 1 + (-0.766 + 0.642i)T \) |
| 47 | \( 1 + (0.939 - 0.342i)T \) |
| 53 | \( 1 + (-0.766 - 0.642i)T \) |
| 59 | \( 1 + (0.939 + 0.342i)T \) |
| 61 | \( 1 + (-0.766 - 0.642i)T \) |
| 67 | \( 1 + (0.939 - 0.342i)T \) |
| 71 | \( 1 + (0.766 - 0.642i)T \) |
| 73 | \( 1 + (-0.173 - 0.984i)T \) |
| 79 | \( 1 + (-0.173 - 0.984i)T \) |
| 83 | \( 1 + (-0.5 + 0.866i)T \) |
| 89 | \( 1 + (-0.173 + 0.984i)T \) |
| 97 | \( 1 + (0.939 + 0.342i)T \) |
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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
−19.91186990558033859663996269924, −19.05760907961501589032446014862, −18.51233574172670594168339360855, −17.41437580007998616946262599415, −16.95357456581919259150805368557, −16.20320970781771574284726096375, −15.65226157060287065633891594347, −14.59108700686743421204576063837, −13.87138951004528185225269914249, −13.435655975931323852076860698786, −12.61800646910755929919501382485, −11.52106753438555660889098399086, −11.19380334333715046960013490205, −10.21827529255022910650999889255, −9.48936897941654088762537391952, −8.768673300960211964775877291, −7.9432286450876320786193607327, −6.972696706530630935056477936388, −6.432176770638256318873929612634, −5.66980523348926519020599638225, −4.382943874752514537416193354978, −3.94197053417943808301057002927, −3.06180690438723617637938069049, −1.9076038300994443227177771848, −0.97887742906511695241656126299,
0.40443623385513619869602253169, 1.88911862428535968011344061255, 2.51394078240832827049328203963, 3.504685994013459801994313940979, 4.39599835957857173250141296254, 5.27347330552323369994674573777, 6.10957932996468527138253032919, 6.73478779058143561932247199769, 7.70659152709733236626705556124, 8.48435112417183408463839308238, 9.32678437392936244783313190452, 9.83393401612379478766439259262, 10.76862397542376658754620976136, 11.60247314926394794521729499807, 12.343057373067697575944078015055, 12.91008086376462170221149391068, 13.64051733372629018956084644326, 14.71625030181254879089127993862, 15.2151277447065733599820906640, 15.81920087551846359319917354032, 16.64970628629264858349638430464, 17.48654959330707941821351622592, 18.16709497283983687046546166227, 18.67294659508480872161379156967, 19.67031002060727691231459941206