L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.104 + 2.23i)5-s + (−1.39 − 2.24i)7-s + 0.999·8-s + (1.98 − 1.02i)10-s + (1.37 + 0.796i)11-s − 0.925·13-s + (−1.24 + 2.33i)14-s + (−0.5 − 0.866i)16-s + (3.42 + 1.97i)17-s + (0.541 − 0.312i)19-s + (−1.88 − 1.20i)20-s − 1.59i·22-s + (3.89 + 6.74i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.0467 + 0.998i)5-s + (−0.528 − 0.848i)7-s + 0.353·8-s + (0.628 − 0.324i)10-s + (0.415 + 0.240i)11-s − 0.256·13-s + (−0.332 + 0.623i)14-s + (−0.125 − 0.216i)16-s + (0.831 + 0.479i)17-s + (0.124 − 0.0717i)19-s + (−0.420 − 0.269i)20-s − 0.339i·22-s + (0.811 + 1.40i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 - 0.518i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.854 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.01671 + 0.284296i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01671 + 0.284296i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.104 - 2.23i)T \) |
| 7 | \( 1 + (1.39 + 2.24i)T \) |
good | 11 | \( 1 + (-1.37 - 0.796i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 0.925T + 13T^{2} \) |
| 17 | \( 1 + (-3.42 - 1.97i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.541 + 0.312i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.89 - 6.74i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 9.34iT - 29T^{2} \) |
| 31 | \( 1 + (-8.94 - 5.16i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.369 + 0.213i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 8.35T + 41T^{2} \) |
| 43 | \( 1 - 6.27iT - 43T^{2} \) |
| 47 | \( 1 + (-2.40 + 1.39i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.67 - 2.90i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.10 + 5.38i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-9.52 + 5.50i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.308 - 0.178i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 9.07iT - 71T^{2} \) |
| 73 | \( 1 + (-3.41 + 5.91i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.52 + 7.83i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 0.809iT - 83T^{2} \) |
| 89 | \( 1 + (-2.00 - 3.47i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 7.87T + 97T^{2} \) |
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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
−10.51976454270867660790467141728, −10.02917383021707277199206208687, −9.213598442072552761336869561825, −7.978552357220069684051029807806, −7.14934962975122290470053132688, −6.48734722808907768256558800830, −5.00929364382458463478376449338, −3.62003488560584837258541109429, −3.08427848801358900413647839411, −1.39329670635780242250180561429,
0.72947719077637058324311092143, 2.54299750051984317215795404186, 4.15273186997758529669648213897, 5.21055735789592536571966329662, 5.98246211698964542068899327891, 6.92000469395597951453633927326, 8.173455064743015547260152418467, 8.623433182316277911538719022102, 9.590234290354133419472977039416, 10.06377899073192092792372072670