L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s − 8-s − 10-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s − 17-s − 19-s + (−0.5 + 0.866i)20-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + 26-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s − 8-s − 10-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s − 17-s − 19-s + (−0.5 + 0.866i)20-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + 26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.2320565332 - 1.316057997i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.2320565332 - 1.316057997i\) |
\(L(1)\) |
\(\approx\) |
\(0.7030567709 - 0.8378704325i\) |
\(L(1)\) |
\(\approx\) |
\(0.7030567709 - 0.8378704325i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.5 + 0.866i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + T \) |
| 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 | \( 1 - T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.5 - 0.866i)T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + (-0.5 + 0.866i)T \) |
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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
−30.704853365679907779928695423628, −29.624302452829281049338010696931, −27.83541967439617885456745399337, −27.15214284241652863830755486331, −25.92589082784004855038372473728, −25.179760217911413720723878453215, −24.00481679828754298678041869275, −23.10002486277109135678581621838, −22.10262789262436026729342848600, −21.327105967759618964136973795519, −19.707959964238709025147895940227, −18.23987379755836539761518778970, −17.73912771477169735369899150496, −16.03378348342793421014649525132, −15.239529176268795551268304869063, −14.49226787911510944669500416921, −13.142292992796741140452942532160, −11.94136037250482192639477705710, −10.7758378771751708528416791984, −8.86420594182160525874561103042, −7.90263254102433317574853118837, −6.6460804957658615088295948707, −5.51828678000151764654658128687, −4.0430384737051644701843642871, −2.66233725267163712183441952150,
0.49788782127968831009992032933, 1.98264787135389382957847533985, 4.078387902583025224995246199110, 4.55678953772056721561370949349, 6.33045376215435625326442922112, 8.162133521255278847591233721719, 9.31262965043258123555196676826, 10.78516402767088491990454892612, 11.61114223896028114724131726138, 12.852471662013827480753092522465, 13.72842859379038253759388272365, 14.93545916594799116929495707542, 16.30218906855167951806245280671, 17.5149683098857924510119945261, 18.89563508072086824429076616278, 19.95345970400574299517484647437, 20.65394588419516048952943867295, 21.59199034838795228927460317929, 23.0088962070407790288686011229, 23.78911483416642691363194694506, 24.51661724080266868780618246638, 26.378879295572039115998929554718, 27.375964820771569561355720483830, 28.28828723841218291268182397409, 29.13513704502863227606308156735