| L(s) = 1 | + 1.70·2-s + 1.92·4-s + 0.406·5-s + 1.57·8-s + 9-s + 0.695·10-s + 0.766·16-s − 1.98·17-s + 1.70·18-s + 0.781·20-s − 0.834·25-s − 0.262·32-s − 3.38·34-s + 1.92·36-s + 1.92·37-s + 0.639·40-s − 1.15·43-s + 0.406·45-s − 1.55·47-s + 49-s − 1.42·50-s − 1.55·61-s − 1.21·64-s + 1.36·67-s − 3.80·68-s − 0.669·71-s + 1.57·72-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.92·4-s + 0.406·5-s + 1.57·8-s + 9-s + 0.695·10-s + 0.766·16-s − 1.98·17-s + 1.70·18-s + 0.781·20-s − 0.834·25-s − 0.262·32-s − 3.38·34-s + 1.92·36-s + 1.92·37-s + 0.639·40-s − 1.15·43-s + 0.406·45-s − 1.55·47-s + 49-s − 1.42·50-s − 1.55·61-s − 1.21·64-s + 1.36·67-s − 3.80·68-s − 0.669·71-s + 1.57·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.474954717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.474954717\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2671 | \( 1+O(T) \) |
| good | 2 | \( 1 - 1.70T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 0.406T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.98T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.92T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.15T + T^{2} \) |
| 47 | \( 1 + 1.55T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.55T + T^{2} \) |
| 67 | \( 1 - 1.36T + T^{2} \) |
| 71 | \( 1 + 0.669T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 1.92T + T^{2} \) |
| 89 | \( 1 + 1.15T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.155365046334752443881053984875, −8.039665190939132343250094589232, −7.10845546584041724360250876629, −6.49508193122702513378004450063, −5.94194002126815159805942981880, −4.86476490297294275743351121676, −4.42776379933131720647822768055, −3.64206234007678049120118093309, −2.51668347345197996916800833406, −1.78253872455005379662515337039,
1.78253872455005379662515337039, 2.51668347345197996916800833406, 3.64206234007678049120118093309, 4.42776379933131720647822768055, 4.86476490297294275743351121676, 5.94194002126815159805942981880, 6.49508193122702513378004450063, 7.10845546584041724360250876629, 8.039665190939132343250094589232, 9.155365046334752443881053984875
