| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.291791789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.291791789\) |
| \(L(1)\) |
\(\approx\) |
\(1.372411122\) |
| \(L(1)\) |
\(\approx\) |
\(1.372411122\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 131 | \( 1 \) |
| 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 | \( 1 + T \) |
| 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 | \( 1 - T \) |
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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
−28.10053949405797751497597689666, −27.48206819897930603478162789249, −26.245709990949587501415297353323, −25.6690023817840441384554054315, −24.68901705426192396383996429606, −24.11122655930628007262105883821, −21.95900116053961313259307203986, −21.00104166473254986579642222624, −20.380788848358368485912569010171, −19.28727450011525264236117569905, −18.15735922290362779426573176589, −17.52780699903546098655249814911, −16.263479424985839767131418457035, −14.95146648574272700902756715717, −14.1946818883451423750950666860, −12.94207615206425877784825218353, −11.324648694991233915382397898150, −10.30229271069095515513599367810, −9.01938987123242348384332874155, −8.597781733817818130966263691234, −7.18696907872269337754588076888, −6.00365414625244325266107823244, −3.99039985431927840912030524596, −2.19320470018265180013840669684, −1.48998515302936687800012193825,
1.48998515302936687800012193825, 2.19320470018265180013840669684, 3.99039985431927840912030524596, 6.00365414625244325266107823244, 7.18696907872269337754588076888, 8.597781733817818130966263691234, 9.01938987123242348384332874155, 10.30229271069095515513599367810, 11.324648694991233915382397898150, 12.94207615206425877784825218353, 14.1946818883451423750950666860, 14.95146648574272700902756715717, 16.263479424985839767131418457035, 17.52780699903546098655249814911, 18.15735922290362779426573176589, 19.28727450011525264236117569905, 20.380788848358368485912569010171, 21.00104166473254986579642222624, 21.95900116053961313259307203986, 24.11122655930628007262105883821, 24.68901705426192396383996429606, 25.6690023817840441384554054315, 26.245709990949587501415297353323, 27.48206819897930603478162789249, 28.10053949405797751497597689666
