L(s) = 1 | + (−0.142 + 0.989i)3-s + (−0.841 − 0.540i)4-s + (0.654 + 0.755i)7-s + (−0.959 − 0.281i)9-s + (0.654 − 0.755i)12-s + (1.25 − 1.45i)13-s + (0.415 + 0.909i)16-s + (0.557 + 1.89i)19-s + (−0.841 + 0.540i)21-s + (−0.654 + 0.755i)25-s + (0.415 − 0.909i)27-s + (−0.142 − 0.989i)28-s + (0.544 + 0.627i)31-s + (0.654 + 0.755i)36-s − 1.30·37-s + ⋯ |
L(s) = 1 | + (−0.142 + 0.989i)3-s + (−0.841 − 0.540i)4-s + (0.654 + 0.755i)7-s + (−0.959 − 0.281i)9-s + (0.654 − 0.755i)12-s + (1.25 − 1.45i)13-s + (0.415 + 0.909i)16-s + (0.557 + 1.89i)19-s + (−0.841 + 0.540i)21-s + (−0.654 + 0.755i)25-s + (0.415 − 0.909i)27-s + (−0.142 − 0.989i)28-s + (0.544 + 0.627i)31-s + (0.654 + 0.755i)36-s − 1.30·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9074109216\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9074109216\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.142 - 0.989i)T \) |
| 7 | \( 1 + (-0.654 - 0.755i)T \) |
| 67 | \( 1 + (-0.654 + 0.755i)T \) |
good | 2 | \( 1 + (0.841 + 0.540i)T^{2} \) |
| 5 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 11 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 13 | \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \) |
| 17 | \( 1 + (0.415 - 0.909i)T^{2} \) |
| 19 | \( 1 + (-0.557 - 1.89i)T + (-0.841 + 0.540i)T^{2} \) |
| 23 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-0.544 - 0.627i)T + (-0.142 + 0.989i)T^{2} \) |
| 37 | \( 1 + 1.30T + T^{2} \) |
| 41 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 43 | \( 1 + (-0.817 - 1.27i)T + (-0.415 + 0.909i)T^{2} \) |
| 47 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 53 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 59 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
| 61 | \( 1 + (-0.118 + 0.258i)T + (-0.654 - 0.755i)T^{2} \) |
| 71 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 73 | \( 1 + (-1.65 - 0.755i)T + (0.654 + 0.755i)T^{2} \) |
| 79 | \( 1 + (1.14 + 0.989i)T + (0.142 + 0.989i)T^{2} \) |
| 83 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 89 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 97 | \( 1 - 0.284T + T^{2} \) |
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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
−9.918947570342871182425181797139, −9.181504059789465519871426831739, −8.326294479133930101411586929821, −8.008400640261710763285599465999, −6.09961540625581400220054914566, −5.61284043889039460550507458339, −5.06086400267791338255890641877, −3.93073882355604430354982915011, −3.21982009106973289087698182176, −1.41930669145409160204404597874,
0.920323668437309628811690524111, 2.27039684019481787884667443255, 3.66686831873195779475968369947, 4.46924219630918436245466158375, 5.38298026253382592057943427027, 6.59560724593486866713159719639, 7.16449348662446643686371193302, 8.012896440954638443741848905292, 8.717218728507969329011085570798, 9.264334896863654628952781729949