| L(s) = 1 | + 2.89·3-s + 4.38·7-s + 5.38·9-s − 2·11-s + 13-s + 5.86·17-s + 0.973·19-s + 12.6·21-s − 7.79·23-s + 6.89·27-s + 0.973·29-s + 1.79·31-s − 5.79·33-s − 0.591·37-s + 2.89·39-s + 4.81·41-s − 4.68·43-s − 0.381·47-s + 12.1·49-s + 16.9·51-s + 7.79·53-s + 2.81·57-s + 0.973·59-s − 0.817·61-s + 23.5·63-s − 1.79·67-s − 22.5·69-s + ⋯ |
| L(s) = 1 | + 1.67·3-s + 1.65·7-s + 1.79·9-s − 0.603·11-s + 0.277·13-s + 1.42·17-s + 0.223·19-s + 2.76·21-s − 1.62·23-s + 1.32·27-s + 0.180·29-s + 0.321·31-s − 1.00·33-s − 0.0972·37-s + 0.463·39-s + 0.752·41-s − 0.714·43-s − 0.0556·47-s + 1.74·49-s + 2.37·51-s + 1.07·53-s + 0.373·57-s + 0.126·59-s − 0.104·61-s + 2.97·63-s − 0.218·67-s − 2.71·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.800782016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.800782016\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 2.89T + 3T^{2} \) |
| 7 | \( 1 - 4.38T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 5.86T + 17T^{2} \) |
| 19 | \( 1 - 0.973T + 19T^{2} \) |
| 23 | \( 1 + 7.79T + 23T^{2} \) |
| 29 | \( 1 - 0.973T + 29T^{2} \) |
| 31 | \( 1 - 1.79T + 31T^{2} \) |
| 37 | \( 1 + 0.591T + 37T^{2} \) |
| 41 | \( 1 - 4.81T + 41T^{2} \) |
| 43 | \( 1 + 4.68T + 43T^{2} \) |
| 47 | \( 1 + 0.381T + 47T^{2} \) |
| 53 | \( 1 - 7.79T + 53T^{2} \) |
| 59 | \( 1 - 0.973T + 59T^{2} \) |
| 61 | \( 1 + 0.817T + 61T^{2} \) |
| 67 | \( 1 + 1.79T + 67T^{2} \) |
| 71 | \( 1 - 3.92T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 6.97T + 83T^{2} \) |
| 89 | \( 1 - 0.973T + 89T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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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
−8.290223833556480646944083657195, −7.70910989327818453648754909383, −7.27308422713902884925491368925, −5.94300873971735319840932821626, −5.18217905131547094084143841125, −4.33113943259900117115475393936, −3.66909513755982831007188551749, −2.75866755916981861086652347420, −1.99579500745269488570612329828, −1.23778228727632873548565789437,
1.23778228727632873548565789437, 1.99579500745269488570612329828, 2.75866755916981861086652347420, 3.66909513755982831007188551749, 4.33113943259900117115475393936, 5.18217905131547094084143841125, 5.94300873971735319840932821626, 7.27308422713902884925491368925, 7.70910989327818453648754909383, 8.290223833556480646944083657195
