L(s) = 1 | + (−0.906 − 0.422i)5-s + (0.422 + 0.906i)11-s + (0.996 − 0.0871i)13-s + (−0.5 + 0.866i)17-s + (0.965 + 0.258i)19-s + (0.984 + 0.173i)23-s + (0.642 + 0.766i)25-s + (0.0871 − 0.996i)29-s + (0.173 − 0.984i)31-s + (0.965 − 0.258i)37-s + (−0.642 + 0.766i)41-s + (0.422 + 0.906i)43-s + (−0.173 − 0.984i)47-s + (0.707 + 0.707i)53-s − i·55-s + ⋯ |
L(s) = 1 | + (−0.906 − 0.422i)5-s + (0.422 + 0.906i)11-s + (0.996 − 0.0871i)13-s + (−0.5 + 0.866i)17-s + (0.965 + 0.258i)19-s + (0.984 + 0.173i)23-s + (0.642 + 0.766i)25-s + (0.0871 − 0.996i)29-s + (0.173 − 0.984i)31-s + (0.965 − 0.258i)37-s + (−0.642 + 0.766i)41-s + (0.422 + 0.906i)43-s + (−0.173 − 0.984i)47-s + (0.707 + 0.707i)53-s − i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.683604600 + 0.4184695451i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.683604600 + 0.4184695451i\) |
\(L(1)\) |
\(\approx\) |
\(1.055677153 + 0.05006555031i\) |
\(L(1)\) |
\(\approx\) |
\(1.055677153 + 0.05006555031i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.906 - 0.422i)T \) |
| 11 | \( 1 + (0.422 + 0.906i)T \) |
| 13 | \( 1 + (0.996 - 0.0871i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.965 + 0.258i)T \) |
| 23 | \( 1 + (0.984 + 0.173i)T \) |
| 29 | \( 1 + (0.0871 - 0.996i)T \) |
| 31 | \( 1 + (0.173 - 0.984i)T \) |
| 37 | \( 1 + (0.965 - 0.258i)T \) |
| 41 | \( 1 + (-0.642 + 0.766i)T \) |
| 43 | \( 1 + (0.422 + 0.906i)T \) |
| 47 | \( 1 + (-0.173 - 0.984i)T \) |
| 53 | \( 1 + (0.707 + 0.707i)T \) |
| 59 | \( 1 + (0.906 + 0.422i)T \) |
| 61 | \( 1 + (-0.819 - 0.573i)T \) |
| 67 | \( 1 + (-0.996 + 0.0871i)T \) |
| 71 | \( 1 + (0.866 + 0.5i)T \) |
| 73 | \( 1 + (0.866 - 0.5i)T \) |
| 79 | \( 1 + (0.766 - 0.642i)T \) |
| 83 | \( 1 + (-0.0871 + 0.996i)T \) |
| 89 | \( 1 + (-0.866 + 0.5i)T \) |
| 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
−17.84701125860912618855616034682, −16.83079081622968422687217115687, −16.168641207991804720195988855884, −15.83269730210442719600312051369, −15.12326555334482880910246407164, −14.3005292565843090089312454192, −13.81368604645981853341674042929, −13.153041135547188171365642514934, −12.21760572878648114900158448937, −11.644333122631137995252973118041, −11.01223324682538392766710351993, −10.7072715787813823872180981537, −9.58002488750615500231918283393, −8.77639553631786454346349203725, −8.49144664360577276868212737434, −7.45856054635295910456138043116, −6.935058808425331667993156265802, −6.32289808601679898893232744917, −5.361025062376784095415406154536, −4.69456094071059805457624295420, −3.74951955272406407823285632170, −3.26192602273855122251396867629, −2.62522169686372054175169494332, −1.28810349046291985200175436972, −0.635557528258386499173425366495,
0.831835345596644738677843140100, 1.480645005581678161633100037, 2.515068807867527844403746160245, 3.48792700423395766644936636532, 4.08203335291591784832546169607, 4.62787776209780304604313225063, 5.530457108966176841754587545926, 6.321759536175697665173026576785, 7.033532789102987009106157788490, 7.85075594623210215936073452788, 8.24238199076847993556087455867, 9.15467281279708357558265345796, 9.62138708533351038841081196818, 10.57930957310364178467079971574, 11.30496445661375637533775007320, 11.74606793127486143173125882532, 12.48054253289303586994015582652, 13.16695750437095116608680239155, 13.62692600623801453530316898371, 14.77818132633598882957109594721, 15.17046206993302121308047341897, 15.65800333643335990686898917127, 16.5658755386956533704797347693, 16.90418317920574725506127192927, 17.82365219226097475910615089957