L(s) = 1 | + (0.809 + 0.587i)3-s − i·7-s + (0.309 + 0.951i)9-s + (0.951 + 0.309i)11-s + (−0.309 − 0.951i)13-s + (−0.587 − 0.809i)17-s + (−0.587 − 0.809i)19-s + (0.587 − 0.809i)21-s + (0.951 + 0.309i)23-s + (−0.309 + 0.951i)27-s + (0.587 − 0.809i)29-s + (−0.809 + 0.587i)31-s + (0.587 + 0.809i)33-s + (−0.309 − 0.951i)37-s + (0.309 − 0.951i)39-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)3-s − i·7-s + (0.309 + 0.951i)9-s + (0.951 + 0.309i)11-s + (−0.309 − 0.951i)13-s + (−0.587 − 0.809i)17-s + (−0.587 − 0.809i)19-s + (0.587 − 0.809i)21-s + (0.951 + 0.309i)23-s + (−0.309 + 0.951i)27-s + (0.587 − 0.809i)29-s + (−0.809 + 0.587i)31-s + (0.587 + 0.809i)33-s + (−0.309 − 0.951i)37-s + (0.309 − 0.951i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.251534831 - 1.081171626i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.251534831 - 1.081171626i\) |
\(L(1)\) |
\(\approx\) |
\(1.424557778 - 0.09762370708i\) |
\(L(1)\) |
\(\approx\) |
\(1.424557778 - 0.09762370708i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.809 + 0.587i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (0.951 + 0.309i)T \) |
| 13 | \( 1 + (-0.309 - 0.951i)T \) |
| 17 | \( 1 + (-0.587 - 0.809i)T \) |
| 19 | \( 1 + (-0.587 - 0.809i)T \) |
| 23 | \( 1 + (0.951 + 0.309i)T \) |
| 29 | \( 1 + (0.587 - 0.809i)T \) |
| 31 | \( 1 + (-0.809 + 0.587i)T \) |
| 37 | \( 1 + (-0.309 - 0.951i)T \) |
| 41 | \( 1 + (-0.309 - 0.951i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.587 - 0.809i)T \) |
| 53 | \( 1 + (-0.809 - 0.587i)T \) |
| 59 | \( 1 + (0.951 - 0.309i)T \) |
| 61 | \( 1 + (-0.951 - 0.309i)T \) |
| 67 | \( 1 + (-0.809 + 0.587i)T \) |
| 71 | \( 1 + (0.809 + 0.587i)T \) |
| 73 | \( 1 + (0.951 + 0.309i)T \) |
| 79 | \( 1 + (0.809 + 0.587i)T \) |
| 83 | \( 1 + (0.809 - 0.587i)T \) |
| 89 | \( 1 + (0.309 - 0.951i)T \) |
| 97 | \( 1 + (0.587 - 0.809i)T \) |
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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
−24.3212371353724449026069629999, −23.76869600719274042982631143987, −22.42399184857229175622347789924, −21.61841746571804470321513025199, −20.84804861080563064290555499509, −19.69607659742357092797038134784, −19.0950617421071004237331213143, −18.468494567355096287891314766805, −17.3502996626757321782659545032, −16.411009836730987766144166068625, −15.09200005761498201215631363615, −14.66072455992453597735493529427, −13.7081633704224505253281407516, −12.59967287610181080056132866935, −12.05118597916925822633807688905, −10.90451532268563641670838718330, −9.36802107181479112091731595989, −8.914211680105157167991569790338, −8.01291715502573008561053125514, −6.71240745300976633222070531939, −6.12625075859434827138640426287, −4.52747939469047356577997581585, −3.416844372024395292747439379370, −2.25961244973493453057325765420, −1.36851379177268183192979884846,
0.62129384838006560688190881823, 2.17468056956061965000760581784, 3.350377124284958566595840264446, 4.26097221845523997485027747258, 5.152332694587012069627951677028, 6.84341328402981452358251835275, 7.51311475105552697953823977317, 8.73966069840770294899683692557, 9.48054476663525500149465394869, 10.4695499002588237987823840837, 11.202544271252825938579409206424, 12.64837891661690064536107240139, 13.555182310885088203176106473931, 14.29633425817093497533289858210, 15.18150811791691618099586180699, 15.96759378849859421874012384462, 17.08239734147739260502051521943, 17.68099451685725147232803441256, 19.196867943539300269057355706677, 19.83138256247949809488404160062, 20.41718480721112675489091708517, 21.32671770676241672140052474682, 22.34263183564035206700797316163, 22.97290882671153087970788960717, 24.196790719343059211537535739876