| L(s) = 1 | − 2-s + 4-s + 0.445·5-s + 0.911·7-s − 8-s − 0.445·10-s + 1.64·11-s − 0.911·14-s + 16-s − 1.50·17-s − 1.60·19-s + 0.445·20-s − 1.64·22-s − 7.38·23-s − 4.80·25-s + 0.911·28-s − 5.24·29-s + 7.34·31-s − 32-s + 1.50·34-s + 0.405·35-s − 2.98·37-s + 1.60·38-s − 0.445·40-s − 5.82·41-s + 2.98·43-s + 1.64·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.199·5-s + 0.344·7-s − 0.353·8-s − 0.140·10-s + 0.495·11-s − 0.243·14-s + 0.250·16-s − 0.365·17-s − 0.367·19-s + 0.0995·20-s − 0.350·22-s − 1.53·23-s − 0.960·25-s + 0.172·28-s − 0.974·29-s + 1.31·31-s − 0.176·32-s + 0.258·34-s + 0.0685·35-s − 0.491·37-s + 0.260·38-s − 0.0703·40-s − 0.909·41-s + 0.455·43-s + 0.247·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 0.445T + 5T^{2} \) |
| 7 | \( 1 - 0.911T + 7T^{2} \) |
| 11 | \( 1 - 1.64T + 11T^{2} \) |
| 17 | \( 1 + 1.50T + 17T^{2} \) |
| 19 | \( 1 + 1.60T + 19T^{2} \) |
| 23 | \( 1 + 7.38T + 23T^{2} \) |
| 29 | \( 1 + 5.24T + 29T^{2} \) |
| 31 | \( 1 - 7.34T + 31T^{2} \) |
| 37 | \( 1 + 2.98T + 37T^{2} \) |
| 41 | \( 1 + 5.82T + 41T^{2} \) |
| 43 | \( 1 - 2.98T + 43T^{2} \) |
| 47 | \( 1 + 7.87T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 6.98T + 61T^{2} \) |
| 67 | \( 1 + 6.81T + 67T^{2} \) |
| 71 | \( 1 + 6.05T + 71T^{2} \) |
| 73 | \( 1 - 8.54T + 73T^{2} \) |
| 79 | \( 1 - 6.73T + 79T^{2} \) |
| 83 | \( 1 + 3.67T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 + 5.77T + 97T^{2} \) |
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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
−8.187350121472182160683098335503, −7.926017243502113929252447575812, −6.78860496745189447055150633712, −6.27875341334983626250156431973, −5.39885398308011029351786626128, −4.36405644664551466604964885112, −3.51571079692847603514504244347, −2.25911194757770128597110828222, −1.53922091242418604742048138197, 0,
1.53922091242418604742048138197, 2.25911194757770128597110828222, 3.51571079692847603514504244347, 4.36405644664551466604964885112, 5.39885398308011029351786626128, 6.27875341334983626250156431973, 6.78860496745189447055150633712, 7.926017243502113929252447575812, 8.187350121472182160683098335503
