L(s) = 1 | + (0.866 + 0.5i)7-s + (−0.5 − 0.866i)11-s + (−0.984 − 0.173i)13-s + (0.342 − 0.939i)17-s + (0.642 − 0.766i)23-s + (0.939 − 0.342i)29-s + (−0.5 + 0.866i)31-s − i·37-s + (−0.173 − 0.984i)41-s + (−0.642 − 0.766i)43-s + (−0.342 − 0.939i)47-s + (0.5 + 0.866i)49-s + (−0.642 + 0.766i)53-s + (0.939 + 0.342i)59-s + (−0.766 − 0.642i)61-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)7-s + (−0.5 − 0.866i)11-s + (−0.984 − 0.173i)13-s + (0.342 − 0.939i)17-s + (0.642 − 0.766i)23-s + (0.939 − 0.342i)29-s + (−0.5 + 0.866i)31-s − i·37-s + (−0.173 − 0.984i)41-s + (−0.642 − 0.766i)43-s + (−0.342 − 0.939i)47-s + (0.5 + 0.866i)49-s + (−0.642 + 0.766i)53-s + (0.939 + 0.342i)59-s + (−0.766 − 0.642i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0457 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0457 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8741982091 - 0.9151649680i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8741982091 - 0.9151649680i\) |
\(L(1)\) |
\(\approx\) |
\(1.016914898 - 0.1672934033i\) |
\(L(1)\) |
\(\approx\) |
\(1.016914898 - 0.1672934033i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 7 | \( 1 + (0.866 + 0.5i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.984 - 0.173i)T \) |
| 17 | \( 1 + (0.342 - 0.939i)T \) |
| 23 | \( 1 + (0.642 - 0.766i)T \) |
| 29 | \( 1 + (0.939 - 0.342i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (-0.173 - 0.984i)T \) |
| 43 | \( 1 + (-0.642 - 0.766i)T \) |
| 47 | \( 1 + (-0.342 - 0.939i)T \) |
| 53 | \( 1 + (-0.642 + 0.766i)T \) |
| 59 | \( 1 + (0.939 + 0.342i)T \) |
| 61 | \( 1 + (-0.766 - 0.642i)T \) |
| 67 | \( 1 + (-0.342 - 0.939i)T \) |
| 71 | \( 1 + (-0.766 + 0.642i)T \) |
| 73 | \( 1 + (-0.984 + 0.173i)T \) |
| 79 | \( 1 + (-0.173 - 0.984i)T \) |
| 83 | \( 1 + (-0.866 - 0.5i)T \) |
| 89 | \( 1 + (0.173 - 0.984i)T \) |
| 97 | \( 1 + (-0.342 + 0.939i)T \) |
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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
−19.705692960677353757164050765384, −19.3614808197245364228726504598, −18.195804894851803561272954286231, −17.688590359907619972993868275988, −17.086735262387748224568466128638, −16.37237910501824701067592386065, −15.356106209173366926017118311116, −14.71348226298360686433219690470, −14.31356617195586005307737211383, −13.17404138817260776280626872681, −12.702001928089160444327065499236, −11.766974722017366984414996033131, −11.11909912485806854205616996186, −10.23879641133050930719865527030, −9.73702150697173363266136453666, −8.72434882947763423919135849818, −7.76218750236965106487508895792, −7.450258007102440621235052461722, −6.4808159275899619006198780537, −5.39095068874838457108034714514, −4.754150209750199525271920493519, −4.06866288178436022130378095141, −2.94903358350978789105826139377, −1.99022159227520769148212883357, −1.22598852484691282266167684891,
0.41969532998251780706029930741, 1.621759951366540125561711977090, 2.654657233272652544408172367250, 3.19487359019961534738709628434, 4.611387391925248138275138496483, 5.07511447349314662128394904644, 5.816933136625295257063902136864, 6.90270361033275755384923284593, 7.61103882586131678691418049067, 8.47785042898584222193176871690, 8.94181334157750277037291613267, 10.07711108647995901505359331598, 10.63425362376986458833310608138, 11.60653419729314211527532769236, 12.03667365375687098106768071011, 12.911312418735546734617084070526, 13.858827071881479250170045985992, 14.35354246564379702473019891034, 15.16642869148754824988178295647, 15.81526327286614983979214693228, 16.67054779681328360897076959005, 17.30544406456130980468119491748, 18.15925925551528621520806376695, 18.67645344765826384179585669546, 19.35593653538230457097504270484