L(s) = 1 | + (−0.741 − 0.670i)2-s + (0.991 − 0.133i)3-s + (0.100 + 0.994i)4-s + (0.964 + 0.264i)5-s + (−0.824 − 0.565i)6-s + (0.860 + 0.509i)7-s + (0.593 − 0.805i)8-s + (0.964 − 0.264i)9-s + (−0.538 − 0.842i)10-s + (0.100 − 0.994i)11-s + (0.231 + 0.972i)12-s + (−0.824 − 0.565i)13-s + (−0.296 − 0.955i)14-s + (0.991 + 0.133i)15-s + (−0.979 + 0.199i)16-s + (−0.296 + 0.955i)17-s + ⋯ |
L(s) = 1 | + (−0.741 − 0.670i)2-s + (0.991 − 0.133i)3-s + (0.100 + 0.994i)4-s + (0.964 + 0.264i)5-s + (−0.824 − 0.565i)6-s + (0.860 + 0.509i)7-s + (0.593 − 0.805i)8-s + (0.964 − 0.264i)9-s + (−0.538 − 0.842i)10-s + (0.100 − 0.994i)11-s + (0.231 + 0.972i)12-s + (−0.824 − 0.565i)13-s + (−0.296 − 0.955i)14-s + (0.991 + 0.133i)15-s + (−0.979 + 0.199i)16-s + (−0.296 + 0.955i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.431529161 - 0.4890982698i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.431529161 - 0.4890982698i\) |
\(L(1)\) |
\(\approx\) |
\(1.210126815 - 0.3160645582i\) |
\(L(1)\) |
\(\approx\) |
\(1.210126815 - 0.3160645582i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 283 | \( 1 \) |
good | 2 | \( 1 + (-0.741 - 0.670i)T \) |
| 3 | \( 1 + (0.991 - 0.133i)T \) |
| 5 | \( 1 + (0.964 + 0.264i)T \) |
| 7 | \( 1 + (0.860 + 0.509i)T \) |
| 11 | \( 1 + (0.100 - 0.994i)T \) |
| 13 | \( 1 + (-0.824 - 0.565i)T \) |
| 17 | \( 1 + (-0.296 + 0.955i)T \) |
| 19 | \( 1 + (-0.824 - 0.565i)T \) |
| 23 | \( 1 + (0.695 - 0.718i)T \) |
| 29 | \( 1 + (-0.892 + 0.451i)T \) |
| 31 | \( 1 + (-0.166 + 0.986i)T \) |
| 37 | \( 1 + (-0.0334 - 0.999i)T \) |
| 41 | \( 1 + (0.593 + 0.805i)T \) |
| 43 | \( 1 + (-0.997 + 0.0667i)T \) |
| 47 | \( 1 + (-0.824 + 0.565i)T \) |
| 53 | \( 1 + (-0.979 - 0.199i)T \) |
| 59 | \( 1 + (0.480 + 0.876i)T \) |
| 61 | \( 1 + (-0.538 + 0.842i)T \) |
| 67 | \( 1 + (0.231 - 0.972i)T \) |
| 71 | \( 1 + (0.100 - 0.994i)T \) |
| 73 | \( 1 + (-0.166 - 0.986i)T \) |
| 79 | \( 1 + (-0.997 - 0.0667i)T \) |
| 83 | \( 1 + (0.920 - 0.390i)T \) |
| 89 | \( 1 + (-0.997 + 0.0667i)T \) |
| 97 | \( 1 + (0.695 - 0.718i)T \) |
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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
−25.69289156905257548455581752385, −24.8612292202536757053008873343, −24.3872216915660383583338113159, −23.27930382609289286771625701839, −21.91493436131518188117273850366, −20.661671564676134415640793150508, −20.4221675594557970051688081482, −19.18808333796259762141730692292, −18.2961845438691171970520632242, −17.30521455405144235247887026018, −16.77229246579548250829505422579, −15.3595268225984573543271669867, −14.61916254596347651904425514394, −13.974713527020016099956126309427, −12.99418113164472785532333910214, −11.3209917244869102927682684077, −9.95707171923918386741348841289, −9.60219649631859884491796341880, −8.56688843076241124327695368198, −7.52069081879558036681406727520, −6.80384060177935614616273801546, −5.17934640734468869373068643988, −4.396500406852027162502523759612, −2.24493625666192782048444779743, −1.60101903951695164555180487277,
1.4832742421598215353864895878, 2.372999657836009176368306207132, 3.24294738658480294155047329566, 4.7802853760276468866507195032, 6.39332098926158278959831528970, 7.635392544360212927948518989954, 8.64784013694368124008048159083, 9.13201883977731500964354643298, 10.385640966263232913842391627922, 11.05905911873331391345894124744, 12.57626981571972916136020265083, 13.18888136162101804572336694994, 14.39754471729589365094298350881, 15.07953351029783794591628791705, 16.57804198339138524814907997524, 17.58978027462479524759048643633, 18.253748549077434848700555771260, 19.17498397781068803149794435596, 19.886548376471948394676258997, 21.12201504254518089163632782480, 21.41282764539081108525750744634, 22.22019471106626595008992445112, 24.173735618374999861144761294105, 24.83330944215474802062816053942, 25.57218287078225787590602908281