| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 13-s − 14-s + 16-s − 17-s − 19-s − 20-s + 23-s + 25-s + 26-s − 28-s − 29-s + 32-s − 34-s + 35-s − 37-s − 38-s − 40-s + 41-s + 43-s + 46-s − 47-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 13-s − 14-s + 16-s − 17-s − 19-s − 20-s + 23-s + 25-s + 26-s − 28-s − 29-s + 32-s − 34-s + 35-s − 37-s − 38-s − 40-s + 41-s + 43-s + 46-s − 47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1023 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1023 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.955993990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.955993990\) |
| \(L(1)\) |
\(\approx\) |
\(1.571563879\) |
| \(L(1)\) |
\(\approx\) |
\(1.571563879\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + T \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.41272818008786118859168341534, −20.71461132328906253709489861281, −19.857728175018612614912917172312, −19.31016720990573222288887410677, −18.60490261422710791320780602986, −17.208412825876423739043700463298, −16.37370677536940241849675984748, −15.715815772817893982479029118204, −15.241080525387054976614359078211, −14.334912249422973394864012122766, −13.11816132202502698667018905415, −12.97003422128086355017721769384, −11.96392023311587335917275594510, −11.040614480609311244205853891728, −10.651445528390005500364587790010, −9.21216527895246978115917852727, −8.35853726668192944634199367395, −7.2308533838576236924099741271, −6.6332053320021172313161953344, −5.803811220093376239036099050613, −4.62596915327846883407268324556, −3.850610717004611129634016477107, −3.229568145040893557724713338237, −2.13934839890412101758369653219, −0.665150687535087266096040330735,
0.665150687535087266096040330735, 2.13934839890412101758369653219, 3.229568145040893557724713338237, 3.850610717004611129634016477107, 4.62596915327846883407268324556, 5.803811220093376239036099050613, 6.6332053320021172313161953344, 7.2308533838576236924099741271, 8.35853726668192944634199367395, 9.21216527895246978115917852727, 10.651445528390005500364587790010, 11.040614480609311244205853891728, 11.96392023311587335917275594510, 12.97003422128086355017721769384, 13.11816132202502698667018905415, 14.334912249422973394864012122766, 15.241080525387054976614359078211, 15.715815772817893982479029118204, 16.37370677536940241849675984748, 17.208412825876423739043700463298, 18.60490261422710791320780602986, 19.31016720990573222288887410677, 19.857728175018612614912917172312, 20.71461132328906253709489861281, 21.41272818008786118859168341534
