L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 13-s + 14-s + 15-s + 16-s + 17-s − 18-s − 19-s − 20-s + 21-s − 22-s + 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯ |
L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 13-s + 14-s + 15-s + 16-s + 17-s − 18-s − 19-s − 20-s + 21-s − 22-s + 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5247737459\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5247737459\) |
\(L(1)\) |
\(\approx\) |
\(0.4790883882\) |
\(L(1)\) |
\(\approx\) |
\(0.4790883882\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| 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 | \( 1 + T \) |
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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
−34.83092840169961121719144213552, −33.4927797237544288292450796231, −32.3101068295994425756839252687, −30.357862018896409935427709001215, −29.464776898064711140571970778951, −28.09335583171493624603559604114, −27.65749009246467670308015460132, −26.323411789686364133656778792272, −24.98922145195416755611140185811, −23.55878453645455334291315462732, −22.64296312368434067990202224085, −20.96389199232680804233909915734, −19.358878795253897913047146670926, −18.74862050556175441281264360145, −17.0873796669794460979342884775, −16.30373940188503475195525493041, −15.24950229092514405157738797122, −12.606143870467470331509577341313, −11.56601380720052741703438312429, −10.44597500315359762114740100317, −8.92663404729950315681373419887, −7.20442176171099355167338900362, −6.1302234797929222268686196454, −3.695326872142827304010572169930, −0.836400774360507167568100327000,
0.836400774360507167568100327000, 3.695326872142827304010572169930, 6.1302234797929222268686196454, 7.20442176171099355167338900362, 8.92663404729950315681373419887, 10.44597500315359762114740100317, 11.56601380720052741703438312429, 12.606143870467470331509577341313, 15.24950229092514405157738797122, 16.30373940188503475195525493041, 17.0873796669794460979342884775, 18.74862050556175441281264360145, 19.358878795253897913047146670926, 20.96389199232680804233909915734, 22.64296312368434067990202224085, 23.55878453645455334291315462732, 24.98922145195416755611140185811, 26.323411789686364133656778792272, 27.65749009246467670308015460132, 28.09335583171493624603559604114, 29.464776898064711140571970778951, 30.357862018896409935427709001215, 32.3101068295994425756839252687, 33.4927797237544288292450796231, 34.83092840169961121719144213552