L(s) = 1 | + (−0.809 − 0.587i)3-s − 0.618·7-s + (−0.587 + 0.190i)13-s + (−0.951 − 1.30i)19-s + (0.500 + 0.363i)21-s + (0.309 − 0.951i)23-s + (−0.309 + 0.951i)27-s + (1.30 + 0.951i)29-s + (0.587 + 0.809i)31-s + (−0.951 + 0.309i)37-s + (0.587 + 0.190i)39-s − 1.61·43-s + (−0.5 − 0.363i)47-s − 0.618·49-s + (−0.587 + 0.809i)53-s + ⋯ |
L(s) = 1 | + (−0.809 − 0.587i)3-s − 0.618·7-s + (−0.587 + 0.190i)13-s + (−0.951 − 1.30i)19-s + (0.500 + 0.363i)21-s + (0.309 − 0.951i)23-s + (−0.309 + 0.951i)27-s + (1.30 + 0.951i)29-s + (0.587 + 0.809i)31-s + (−0.951 + 0.309i)37-s + (0.587 + 0.190i)39-s − 1.61·43-s + (−0.5 − 0.363i)47-s − 0.618·49-s + (−0.587 + 0.809i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.001349061221\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.001349061221\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + 0.618T + T^{2} \) |
| 11 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + 1.61T + T^{2} \) |
| 47 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 71 | \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.245715467393627445374602541742, −7.01230645882854777620421640941, −6.70312511570715223156163778469, −6.27096509829528386896867986702, −5.05999693966374599208256923731, −4.70186060927938040702545539008, −3.36669611713169556130848060550, −2.60929555515067137449739993231, −1.32194223689711514608347402316, −0.000846644362155833575642995893,
1.78685198367493889744002326325, 2.93147685622055406538461647097, 3.84983018069211720660885462636, 4.67903373156426047076434956911, 5.31729060359090405236493951837, 6.18437599687721968710701797649, 6.57005145961567404977108999231, 7.77482580826659222579334553655, 8.203017985838717898609751914291, 9.264185172255693344713925188055