L(s) = 1 | + 3.23·5-s + 1.23·7-s − 0.763·11-s + 3·13-s − 5.23·17-s + 2·19-s + 23-s + 5.47·25-s + 3·29-s + 6.70·31-s + 4.00·35-s + 3.23·37-s − 5.47·41-s + 2.23·47-s − 5.47·49-s + 8.47·53-s − 2.47·55-s − 2.47·59-s + 10.9·61-s + 9.70·65-s + 7.23·67-s + 7.76·71-s + 15.4·73-s − 0.944·77-s − 6.94·79-s − 13.2·83-s − 16.9·85-s + ⋯ |
L(s) = 1 | + 1.44·5-s + 0.467·7-s − 0.230·11-s + 0.832·13-s − 1.26·17-s + 0.458·19-s + 0.208·23-s + 1.09·25-s + 0.557·29-s + 1.20·31-s + 0.676·35-s + 0.532·37-s − 0.854·41-s + 0.326·47-s − 0.781·49-s + 1.16·53-s − 0.333·55-s − 0.321·59-s + 1.40·61-s + 1.20·65-s + 0.884·67-s + 0.921·71-s + 1.81·73-s − 0.107·77-s − 0.781·79-s − 1.45·83-s − 1.83·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.772695111\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.772695111\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 + 0.763T + 11T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 6.70T + 31T^{2} \) |
| 37 | \( 1 - 3.23T + 37T^{2} \) |
| 41 | \( 1 + 5.47T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 2.23T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 + 2.47T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 7.23T + 67T^{2} \) |
| 71 | \( 1 - 7.76T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 + 6.94T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 - 1.52T + 89T^{2} \) |
| 97 | \( 1 - 4.29T + 97T^{2} \) |
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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
−8.591534390928548757234011838919, −8.129268070565906120717895357667, −6.85858974195832455577073939946, −6.44479739570541910052410123862, −5.57640241803011706020389774711, −4.96933744408047972299717843674, −4.03226673433012479853632874276, −2.79410498399196885244730163888, −2.05175877783164207980030865695, −1.06212686368271301802238229240,
1.06212686368271301802238229240, 2.05175877783164207980030865695, 2.79410498399196885244730163888, 4.03226673433012479853632874276, 4.96933744408047972299717843674, 5.57640241803011706020389774711, 6.44479739570541910052410123862, 6.85858974195832455577073939946, 8.129268070565906120717895357667, 8.591534390928548757234011838919