L(s) = 1 | + 1.73i·3-s + (3.62 − 2.09i)5-s + (1 − 2.44i)7-s − 2.99·9-s + (−1.5 − 0.866i)11-s + (0.621 + 0.358i)13-s + (3.62 + 6.27i)15-s + (5.74 − 3.31i)17-s + (0.5 − 0.866i)19-s + (4.24 + 1.73i)21-s + (−6.62 + 3.82i)23-s + (6.24 − 10.8i)25-s − 5.19i·27-s + (−3.62 − 6.27i)29-s + 4·31-s + ⋯ |
L(s) = 1 | + 0.999i·3-s + (1.61 − 0.935i)5-s + (0.377 − 0.925i)7-s − 0.999·9-s + (−0.452 − 0.261i)11-s + (0.172 + 0.0994i)13-s + (0.935 + 1.61i)15-s + (1.39 − 0.804i)17-s + (0.114 − 0.198i)19-s + (0.925 + 0.377i)21-s + (−1.38 + 0.797i)23-s + (1.24 − 2.16i)25-s − 0.999i·27-s + (−0.672 − 1.16i)29-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.109284900\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.109284900\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 7 | \( 1 + (-1 + 2.44i)T \) |
good | 5 | \( 1 + (-3.62 + 2.09i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 + 0.866i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.621 - 0.358i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-5.74 + 3.31i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.62 - 3.82i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.62 + 6.27i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (2.62 - 4.54i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.257 + 0.148i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.74 - 1.58i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 + (-3.62 - 6.27i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 6.33iT - 71T^{2} \) |
| 73 | \( 1 + (7.5 - 4.33i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 11.8iT - 79T^{2} \) |
| 83 | \( 1 + (-2.74 - 4.75i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-11.2 - 6.48i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.74 + 1.58i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.08335042889671983281943473330, −9.340870506179773064652001142682, −8.436865887391115976394540286992, −7.60234295603218662908612914851, −6.11016725199287614420477759324, −5.49818713735776871252522519905, −4.80673353384998062447147785189, −3.79764254009963765980089672984, −2.44911386473045651008386245777, −1.03476894378478545090356077866,
1.67303549482966478457324910943, 2.28781298206815606137789073820, 3.28804122939699773957817524262, 5.35012750745835212495512374283, 5.79135299973560000147760190006, 6.46285195571087344142202047503, 7.43650563563790192929922512307, 8.298521252766716272703270849366, 9.141681839403262118325085415211, 10.17576945607908359195822907692