L(s) = 1 | − 2i·7-s − 1.41·11-s + 13-s − 1.41i·17-s − 25-s + 1.41i·29-s − 1.41·47-s − 3·49-s − 1.41i·53-s − 1.41·59-s − 2i·67-s + 1.41·71-s + 2.82i·77-s + 1.41·83-s − 2i·91-s + ⋯ |
L(s) = 1 | − 2i·7-s − 1.41·11-s + 13-s − 1.41i·17-s − 25-s + 1.41i·29-s − 1.41·47-s − 3·49-s − 1.41i·53-s − 1.41·59-s − 2i·67-s + 1.41·71-s + 2.82i·77-s + 1.41·83-s − 2i·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9073179420\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9073179420\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + 2iT - T^{2} \) |
| 11 | \( 1 + 1.41T + T^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + 1.41T + T^{2} \) |
| 53 | \( 1 + 1.41iT - T^{2} \) |
| 59 | \( 1 + 1.41T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + 2iT - T^{2} \) |
| 71 | \( 1 - 1.41T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - 1.41T + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 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
−8.089007236226471041765851554525, −7.80199081751594944198134258925, −7.01003096028855223638669091304, −6.41820869267192493544048151116, −5.21063137483275099168203687998, −4.72516415845926324803587653670, −3.67461667684720974056528045528, −3.12747041926156170629589083624, −1.71997506190771126393735826332, −0.49703993500675149397681882914,
1.79049215104578097119088190204, 2.49891886504529678394497968118, 3.37128124558202806589129757407, 4.44749028957424214909209794350, 5.46669440232052679027970366996, 5.88497423548763766168385900960, 6.42008448299702065961655562028, 7.901784510883224051711965992428, 8.128003043767961413575405066865, 8.833239510556776321798657737592