L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 11-s + 12-s + 13-s − 15-s + 16-s − 17-s − 18-s − 20-s − 22-s + 23-s − 24-s + 25-s − 26-s + 27-s − 29-s + 30-s + 31-s − 32-s + 33-s + ⋯ |
L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 11-s + 12-s + 13-s − 15-s + 16-s − 17-s − 18-s − 20-s − 22-s + 23-s − 24-s + 25-s − 26-s + 27-s − 29-s + 30-s + 31-s − 32-s + 33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9084590317\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9084590317\) |
\(L(1)\) |
\(\approx\) |
\(0.8936883087\) |
\(L(1)\) |
\(\approx\) |
\(0.8936883087\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 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
−28.24758837664794843697727615596, −27.46824532055970711680240660253, −26.66137132762555261058646946522, −25.873482625399781992545440174899, −24.77625639764276167769021135215, −24.11795044361899509082713214847, −22.65193510260904049207325830085, −21.08112790359596772335582481385, −20.28573769920505928756350109642, −19.39531025590113440579401678424, −18.846640952188759442978194100199, −17.56045729012685395588945963231, −16.19422064040168239364255689780, −15.45340298751968508614649748391, −14.52780433872581024357404687792, −13.00238459337027161207626894526, −11.671390658087327129590431380805, −10.72220816155185961236418829983, −9.18601306679871263617429376743, −8.622107034578946389255971755361, −7.50667577710357321372913226853, −6.537290780686861531090829493660, −4.151323040637174249382431018531, −3.02676951049389218311006396684, −1.39047300131328585995249715062,
1.39047300131328585995249715062, 3.02676951049389218311006396684, 4.151323040637174249382431018531, 6.537290780686861531090829493660, 7.50667577710357321372913226853, 8.622107034578946389255971755361, 9.18601306679871263617429376743, 10.72220816155185961236418829983, 11.671390658087327129590431380805, 13.00238459337027161207626894526, 14.52780433872581024357404687792, 15.45340298751968508614649748391, 16.19422064040168239364255689780, 17.56045729012685395588945963231, 18.846640952188759442978194100199, 19.39531025590113440579401678424, 20.28573769920505928756350109642, 21.08112790359596772335582481385, 22.65193510260904049207325830085, 24.11795044361899509082713214847, 24.77625639764276167769021135215, 25.873482625399781992545440174899, 26.66137132762555261058646946522, 27.46824532055970711680240660253, 28.24758837664794843697727615596