L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s − 7-s + (−0.5 + 0.866i)9-s − 11-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)21-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + 27-s + (0.5 − 0.866i)29-s + 31-s + (0.5 + 0.866i)33-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s − 7-s + (−0.5 + 0.866i)9-s − 11-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)21-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + 27-s + (0.5 − 0.866i)29-s + 31-s + (0.5 + 0.866i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1513100478 - 0.4711510755i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1513100478 - 0.4711510755i\) |
\(L(1)\) |
\(\approx\) |
\(0.5444585745 - 0.3554927930i\) |
\(L(1)\) |
\(\approx\) |
\(0.5444585745 - 0.3554927930i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + T \) |
| 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 | \( 1 + (0.5 - 0.866i)T \) |
| 59 | \( 1 + (-0.5 - 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 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 | \( 1 + (0.5 + 0.866i)T \) |
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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
−31.7612094107870519046522065940, −30.88907762195413766939418563356, −29.33340741935054888222392611183, −28.6154775272682322051572108980, −27.42585564007553546541564237415, −26.20576025758494651494936786084, −25.956065664355465041186329675565, −23.74014861695812710581901882900, −23.02462314323435186793988287226, −22.03149539372091202591994665187, −21.09570815109919306970357185341, −19.606763176519492967335996780085, −18.599214968113827970794717877471, −17.254153142848586422501757409837, −15.85937534171682304035638596682, −15.4487651538860355056711621140, −13.89513595701432375219917248887, −12.33300278153568286834102221757, −10.99746765501690092284644137077, −10.24687219894998237649489227321, −8.8581221353773641045339907238, −7.019664629035192349384364112658, −5.87757034690246130943089346732, −4.15290600150176723047257843734, −3.00082467881282824216785862619,
0.58643777676603813148578083397, 2.80722110358583131167560499354, 4.84961988064021471629710131160, 6.14007255244981134512812111322, 7.533294131612715784735581358914, 8.66032633146557685740066672257, 10.3684854346823042521354082626, 11.783820688072526194670827910777, 12.87438834997580717123564238752, 13.438080816455064444966024837614, 15.61374586037738500121590160878, 16.37203995166865835259410108279, 17.65724732039485731661817504769, 18.788972818306042386113536472942, 19.78498907847391683258141524432, 20.87123947515133948945018891343, 22.7064726909371224521727937056, 23.16953424639576405869516568038, 24.44243922779136480881675444591, 25.212662284167302064462573807574, 26.593525980923935541086053795573, 28.09368364796693274089527945158, 28.731284905853535931558904026039, 29.65968366310593268952569635033, 30.96453725105437204053826179350