L(s) = 1 | − 0.323·2-s − 1.89·4-s + 3.36·5-s + 3.36·7-s + 1.25·8-s − 1.08·10-s − 1.65·11-s + 4.48·13-s − 1.08·14-s + 3.38·16-s + 0.370·17-s − 3.44·19-s − 6.37·20-s + 0.536·22-s − 3.00·23-s + 6.31·25-s − 1.45·26-s − 6.37·28-s − 0.548·29-s + 2.39·31-s − 3.61·32-s − 0.119·34-s + 11.3·35-s − 2.38·37-s + 1.11·38-s + 4.23·40-s + 4.88·41-s + ⋯ |
L(s) = 1 | − 0.228·2-s − 0.947·4-s + 1.50·5-s + 1.27·7-s + 0.445·8-s − 0.344·10-s − 0.500·11-s + 1.24·13-s − 0.290·14-s + 0.845·16-s + 0.0897·17-s − 0.790·19-s − 1.42·20-s + 0.114·22-s − 0.625·23-s + 1.26·25-s − 0.284·26-s − 1.20·28-s − 0.101·29-s + 0.430·31-s − 0.638·32-s − 0.0205·34-s + 1.91·35-s − 0.391·37-s + 0.180·38-s + 0.670·40-s + 0.762·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.425027154\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.425027154\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 223 | \( 1 - T \) |
good | 2 | \( 1 + 0.323T + 2T^{2} \) |
| 5 | \( 1 - 3.36T + 5T^{2} \) |
| 7 | \( 1 - 3.36T + 7T^{2} \) |
| 11 | \( 1 + 1.65T + 11T^{2} \) |
| 13 | \( 1 - 4.48T + 13T^{2} \) |
| 17 | \( 1 - 0.370T + 17T^{2} \) |
| 19 | \( 1 + 3.44T + 19T^{2} \) |
| 23 | \( 1 + 3.00T + 23T^{2} \) |
| 29 | \( 1 + 0.548T + 29T^{2} \) |
| 31 | \( 1 - 2.39T + 31T^{2} \) |
| 37 | \( 1 + 2.38T + 37T^{2} \) |
| 41 | \( 1 - 4.88T + 41T^{2} \) |
| 43 | \( 1 - 8.31T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 7.04T + 53T^{2} \) |
| 59 | \( 1 + 2.72T + 59T^{2} \) |
| 61 | \( 1 + 0.972T + 61T^{2} \) |
| 67 | \( 1 + 7.10T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 5.65T + 73T^{2} \) |
| 79 | \( 1 - 7.20T + 79T^{2} \) |
| 83 | \( 1 - 5.70T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 3.30T + 97T^{2} \) |
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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
−8.221489869001578358576453247663, −7.63348296483674861548500224744, −6.48943974800293855226493947039, −5.74476492900547490127636024148, −5.35792099750107917885103468534, −4.50246320703259346044904581927, −3.85331386529853152659511082259, −2.51067316790513673353993642412, −1.73452054768724283728299566653, −0.923432101294479770349458908882,
0.923432101294479770349458908882, 1.73452054768724283728299566653, 2.51067316790513673353993642412, 3.85331386529853152659511082259, 4.50246320703259346044904581927, 5.35792099750107917885103468534, 5.74476492900547490127636024148, 6.48943974800293855226493947039, 7.63348296483674861548500224744, 8.221489869001578358576453247663