L(s) = 1 | + (−0.191 + 0.981i)2-s + (−0.972 + 0.233i)3-s + (−0.926 − 0.375i)4-s + (−0.977 + 0.209i)5-s + (−0.0434 − 0.999i)6-s + (0.503 + 0.863i)7-s + (0.545 − 0.837i)8-s + (0.890 − 0.454i)9-s + (−0.0186 − 0.999i)10-s + (−0.860 + 0.508i)11-s + (0.988 + 0.148i)12-s + (−0.743 − 0.668i)13-s + (−0.944 + 0.329i)14-s + (0.901 − 0.432i)15-s + (0.717 + 0.696i)16-s + (−0.986 − 0.160i)17-s + ⋯ |
L(s) = 1 | + (−0.191 + 0.981i)2-s + (−0.972 + 0.233i)3-s + (−0.926 − 0.375i)4-s + (−0.977 + 0.209i)5-s + (−0.0434 − 0.999i)6-s + (0.503 + 0.863i)7-s + (0.545 − 0.837i)8-s + (0.890 − 0.454i)9-s + (−0.0186 − 0.999i)10-s + (−0.860 + 0.508i)11-s + (0.988 + 0.148i)12-s + (−0.743 − 0.668i)13-s + (−0.944 + 0.329i)14-s + (0.901 − 0.432i)15-s + (0.717 + 0.696i)16-s + (−0.986 − 0.160i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4123958656 + 0.2146757691i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4123958656 + 0.2146757691i\) |
\(L(1)\) |
\(\approx\) |
\(0.4484776340 + 0.2604088989i\) |
\(L(1)\) |
\(\approx\) |
\(0.4484776340 + 0.2604088989i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (-0.191 + 0.981i)T \) |
| 3 | \( 1 + (-0.972 + 0.233i)T \) |
| 5 | \( 1 + (-0.977 + 0.209i)T \) |
| 7 | \( 1 + (0.503 + 0.863i)T \) |
| 11 | \( 1 + (-0.860 + 0.508i)T \) |
| 13 | \( 1 + (-0.743 - 0.668i)T \) |
| 17 | \( 1 + (-0.986 - 0.160i)T \) |
| 19 | \( 1 + (-0.0929 - 0.995i)T \) |
| 29 | \( 1 + (0.969 - 0.245i)T \) |
| 31 | \( 1 + (0.0310 - 0.999i)T \) |
| 37 | \( 1 + (0.980 - 0.197i)T \) |
| 41 | \( 1 + (0.940 - 0.340i)T \) |
| 43 | \( 1 + (-0.834 - 0.551i)T \) |
| 47 | \( 1 + (-0.775 - 0.631i)T \) |
| 53 | \( 1 + (-0.0186 + 0.999i)T \) |
| 59 | \( 1 + (-0.616 + 0.787i)T \) |
| 61 | \( 1 + (0.798 + 0.601i)T \) |
| 67 | \( 1 + (-0.449 + 0.893i)T \) |
| 71 | \( 1 + (0.0806 + 0.996i)T \) |
| 73 | \( 1 + (0.969 + 0.245i)T \) |
| 79 | \( 1 + (0.645 + 0.763i)T \) |
| 83 | \( 1 + (0.997 + 0.0744i)T \) |
| 89 | \( 1 + (-0.166 - 0.985i)T \) |
| 97 | \( 1 + (-0.998 - 0.0620i)T \) |
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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
−23.43017504420924517858367107371, −22.54765600582524766801667597110, −21.572799806771467395339819778593, −20.91727713973774918834999586933, −19.820752967604193032548559926272, −19.28887352804360565560744468819, −18.300452299558611467407998547740, −17.64583620768020557710878513108, −16.6380667212309797806756988284, −16.13618753787497864768138331054, −14.67536783235410647666850466827, −13.611426949637457030709785517542, −12.73045410206071692822392377694, −12.01784665818420346568422615, −11.126269769571121761629853349748, −10.74349804389884888422757998612, −9.70842697738956934205957758377, −8.25621777711448535551734471951, −7.723095832359272965035924556819, −6.56651500087060532700720714022, −4.889123485126490985830479869326, −4.579327384842654050906063332600, −3.41823925688289774220364286513, −1.90373336777692419777153516868, −0.72609389233992414725832983847,
0.50173245522415591321860137065, 2.557891552070850752880272774990, 4.33196978864629899840341006629, 4.85750745955236313347971920533, 5.73705736261922238958656614513, 6.85802644784165902618159114752, 7.59765761794983225301734618352, 8.502164372286460232653532819582, 9.60740768674033359823835668473, 10.60606026201633093786364952216, 11.47152342999322880310604634626, 12.42946252137352630020275901296, 13.21628834699390921759777269686, 14.7823886537921982482812578529, 15.43844057403768083536941016493, 15.67109609903450143350602144172, 16.79957805116503647102112494695, 17.85094979633922780370343612664, 18.08383739385977083263030705727, 19.11193321421559485628688897703, 20.1252353025589349343000657068, 21.496428763081106603483872052890, 22.23460302786528466800747215311, 22.87829223825192179679383381195, 23.705577751921437949185705388677