| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.230895546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.230895546\) |
| \(L(1)\) |
\(\approx\) |
\(1.075468516\) |
| \(L(1)\) |
\(\approx\) |
\(1.075468516\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 229 | \( 1 \) |
| 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
−26.18949138511800975866004950295, −25.59769360316515237976889424387, −24.90310001929554511217499706493, −24.176185080088642661483924236213, −22.288549690254344183810255673557, −21.59132033966322428032182789191, −20.37760660657860043486157939872, −19.82757201815480202805369721914, −18.900388335640810399316951521294, −18.0902043555892238933448636873, −16.86567472546333718741589320853, −16.24396807367554434993470683239, −14.88742431338322266792510315610, −14.16919909262047839026186084536, −12.92326293376044943540170690917, −11.943674560809925112465665005834, −10.21793024564495532964696993855, −9.59607587149019623464214149570, −9.132438714426825889537937101772, −7.695169240638416819110578034165, −6.830187081529140607621334276032, −5.69340019193448016576079332556, −3.59121195248544653845483107529, −2.55775806419469608784620593553, −1.43154855808135505725903151598,
1.43154855808135505725903151598, 2.55775806419469608784620593553, 3.59121195248544653845483107529, 5.69340019193448016576079332556, 6.830187081529140607621334276032, 7.695169240638416819110578034165, 9.132438714426825889537937101772, 9.59607587149019623464214149570, 10.21793024564495532964696993855, 11.943674560809925112465665005834, 12.92326293376044943540170690917, 14.16919909262047839026186084536, 14.88742431338322266792510315610, 16.24396807367554434993470683239, 16.86567472546333718741589320853, 18.0902043555892238933448636873, 18.900388335640810399316951521294, 19.82757201815480202805369721914, 20.37760660657860043486157939872, 21.59132033966322428032182789191, 22.288549690254344183810255673557, 24.176185080088642661483924236213, 24.90310001929554511217499706493, 25.59769360316515237976889424387, 26.18949138511800975866004950295
