| L(s) = 1 | + 0.732·2-s − 1.46·4-s + 5-s + 7-s − 2.53·8-s + 0.732·10-s + 11-s − 4.19·13-s + 0.732·14-s + 1.07·16-s + 6.46·17-s − 1.53·19-s − 1.46·20-s + 0.732·22-s − 8.46·23-s + 25-s − 3.07·26-s − 1.46·28-s − 5.73·29-s − 5.26·31-s + 5.85·32-s + 4.73·34-s + 35-s − 4.19·37-s − 1.12·38-s − 2.53·40-s + 7.66·41-s + ⋯ |
| L(s) = 1 | + 0.517·2-s − 0.732·4-s + 0.447·5-s + 0.377·7-s − 0.896·8-s + 0.231·10-s + 0.301·11-s − 1.16·13-s + 0.195·14-s + 0.267·16-s + 1.56·17-s − 0.352·19-s − 0.327·20-s + 0.156·22-s − 1.76·23-s + 0.200·25-s − 0.602·26-s − 0.276·28-s − 1.06·29-s − 0.946·31-s + 1.03·32-s + 0.811·34-s + 0.169·35-s − 0.689·37-s − 0.182·38-s − 0.400·40-s + 1.19·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 0.732T + 2T^{2} \) |
| 13 | \( 1 + 4.19T + 13T^{2} \) |
| 17 | \( 1 - 6.46T + 17T^{2} \) |
| 19 | \( 1 + 1.53T + 19T^{2} \) |
| 23 | \( 1 + 8.46T + 23T^{2} \) |
| 29 | \( 1 + 5.73T + 29T^{2} \) |
| 31 | \( 1 + 5.26T + 31T^{2} \) |
| 37 | \( 1 + 4.19T + 37T^{2} \) |
| 41 | \( 1 - 7.66T + 41T^{2} \) |
| 43 | \( 1 - 1.73T + 43T^{2} \) |
| 47 | \( 1 + 1.26T + 47T^{2} \) |
| 53 | \( 1 + 4.46T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 - 4.92T + 67T^{2} \) |
| 71 | \( 1 + 7.66T + 71T^{2} \) |
| 73 | \( 1 - 8.39T + 73T^{2} \) |
| 79 | \( 1 - 3.66T + 79T^{2} \) |
| 83 | \( 1 + 2.46T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 2.26T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.065407776510532093839153858023, −7.66189841584485728434900570870, −6.57357558397370905276555626252, −5.55682464769454182896073152541, −5.39869346218672891639910679746, −4.29907252782526253377472981168, −3.70642906090214743588579595325, −2.62687083849799696594362132525, −1.54297570215907051646939419224, 0,
1.54297570215907051646939419224, 2.62687083849799696594362132525, 3.70642906090214743588579595325, 4.29907252782526253377472981168, 5.39869346218672891639910679746, 5.55682464769454182896073152541, 6.57357558397370905276555626252, 7.66189841584485728434900570870, 8.065407776510532093839153858023
