L(s) = 1 | + (0.950 + 0.548i)2-s + (0.328 + 0.189i)3-s + (−0.397 − 0.689i)4-s + (0.208 + 0.360i)6-s − 1.89i·7-s − 3.06i·8-s + (−1.42 − 2.47i)9-s + 0.134·11-s − 0.301i·12-s + (3.04 − 1.75i)13-s + (1.03 − 1.79i)14-s + (0.887 − 1.53i)16-s + (−1.43 − 0.830i)17-s − 3.13i·18-s + (−2.10 + 3.81i)19-s + ⋯ |
L(s) = 1 | + (0.672 + 0.388i)2-s + (0.189 + 0.109i)3-s + (−0.198 − 0.344i)4-s + (0.0849 + 0.147i)6-s − 0.715i·7-s − 1.08i·8-s + (−0.476 − 0.824i)9-s + 0.0405·11-s − 0.0871i·12-s + (0.843 − 0.487i)13-s + (0.277 − 0.480i)14-s + (0.221 − 0.384i)16-s + (−0.348 − 0.201i)17-s − 0.738i·18-s + (−0.483 + 0.875i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.641 + 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.641 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.66841 - 0.780302i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.66841 - 0.780302i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (2.10 - 3.81i)T \) |
good | 2 | \( 1 + (-0.950 - 0.548i)T + (1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.328 - 0.189i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + 1.89iT - 7T^{2} \) |
| 11 | \( 1 - 0.134T + 11T^{2} \) |
| 13 | \( 1 + (-3.04 + 1.75i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.43 + 0.830i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-4.65 + 2.68i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.48 - 4.30i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.56T + 31T^{2} \) |
| 37 | \( 1 - 1.69iT - 37T^{2} \) |
| 41 | \( 1 + (5.31 - 9.20i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (7.36 + 4.25i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-9.62 + 5.55i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.229 - 0.132i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.44 - 5.97i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.58 + 7.94i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.55 + 1.47i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.664 - 1.15i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.49 - 3.17i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.733 - 1.27i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 7.44iT - 83T^{2} \) |
| 89 | \( 1 + (-4.86 - 8.43i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-15.1 - 8.73i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.70608768533886866738309526131, −10.11648909184908974563618394687, −9.036296312807081771200290632612, −8.237055025870552864850413967260, −6.81077956087569311739697384189, −6.27806233404564315386034463575, −5.14920808202233975113753020402, −4.10508382166596380410414596263, −3.21677857207363501647950811476, −0.961766580424582837667672865557,
2.14958811639758940628369202044, 3.07845209317665553219980119779, 4.35717122339405510658071405974, 5.23433738816325047608459755785, 6.30154810092315487270352529006, 7.61751127808572066601245525272, 8.656442998493318326474877939172, 8.968970211305603800070720163258, 10.55675427312163725961308126178, 11.37259430126930388864555013184