L(s) = 1 | − 2-s + 4-s + (−1.82 + 3.15i)5-s + (1.32 − 2.29i)7-s − 8-s + (1.82 − 3.15i)10-s + (1.82 + 3.15i)11-s + (2.32 + 4.02i)13-s + (−1.32 + 2.29i)14-s + 16-s + (1.82 − 3.15i)17-s + (−1 − 1.73i)19-s + (−1.82 + 3.15i)20-s + (−1.82 − 3.15i)22-s + (−0.645 + 1.11i)23-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + (−0.815 + 1.41i)5-s + (0.499 − 0.866i)7-s − 0.353·8-s + (0.576 − 0.998i)10-s + (0.549 + 0.951i)11-s + (0.644 + 1.11i)13-s + (−0.353 + 0.612i)14-s + 0.250·16-s + (0.442 − 0.765i)17-s + (−0.229 − 0.397i)19-s + (−0.407 + 0.705i)20-s + (−0.388 − 0.673i)22-s + (−0.134 + 0.233i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9301802103\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9301802103\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.32 + 2.29i)T \) |
good | 5 | \( 1 + (1.82 - 3.15i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.82 - 3.15i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.32 - 4.02i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.82 + 3.15i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.645 - 1.11i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.17 + 2.03i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4.64T + 31T^{2} \) |
| 37 | \( 1 + (-5.96 - 10.3i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.46 - 9.47i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 4.93T + 47T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 8.35T + 59T^{2} \) |
| 61 | \( 1 - 7.35T + 61T^{2} \) |
| 67 | \( 1 + 2.29T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 + (5.29 - 9.16i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 + (6.64 - 11.5i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.46 - 4.27i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.79 - 11.7i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.00144954777303215751117852285, −9.446895078459894345188396424465, −8.200637184829053414971903548111, −7.56121047308883670001713893469, −6.86375258375659370181209398961, −6.41645055344304494215405708390, −4.63131605175908964682872831481, −3.85458270703375174296988063953, −2.76740580576426908453101366868, −1.40616753698785388902155265764,
0.56833349665882021691322437837, 1.68557795047085883458816270455, 3.32019966378888857729744195819, 4.26750867928268588966002406050, 5.60563578632556473619879848351, 5.89285723440028100597334001316, 7.54587971847128805906498014164, 8.148509875515797150452065514163, 8.807394320568037103241203113194, 9.077355892155555736557551218098