L(s) = 1 | − 2.41i·2-s + (−3.31 + 4.00i)3-s + 2.16·4-s + (−5.35 − 9.28i)5-s + (9.67 + 8.00i)6-s + (3.86 − 18.1i)7-s − 24.5i·8-s + (−5.05 − 26.5i)9-s + (−22.4 + 12.9i)10-s + (−22.6 − 13.0i)11-s + (−7.15 + 8.64i)12-s + (29.8 + 17.2i)13-s + (−43.7 − 9.33i)14-s + (54.9 + 9.29i)15-s − 42.0·16-s + (−2.29 − 3.98i)17-s + ⋯ |
L(s) = 1 | − 0.854i·2-s + (−0.637 + 0.770i)3-s + 0.270·4-s + (−0.479 − 0.830i)5-s + (0.658 + 0.544i)6-s + (0.208 − 0.978i)7-s − 1.08i·8-s + (−0.187 − 0.982i)9-s + (−0.709 + 0.409i)10-s + (−0.620 − 0.358i)11-s + (−0.172 + 0.208i)12-s + (0.637 + 0.368i)13-s + (−0.835 − 0.178i)14-s + (0.945 + 0.159i)15-s − 0.657·16-s + (−0.0327 − 0.0567i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.567450 - 0.953810i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.567450 - 0.953810i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (3.31 - 4.00i)T \) |
| 7 | \( 1 + (-3.86 + 18.1i)T \) |
good | 2 | \( 1 + 2.41iT - 8T^{2} \) |
| 5 | \( 1 + (5.35 + 9.28i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (22.6 + 13.0i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-29.8 - 17.2i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (2.29 + 3.98i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-6.74 - 3.89i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (33.9 - 19.5i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-210. + 121. i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 311. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (28.3 - 49.0i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (148. - 256. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (60.7 + 105. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 - 225.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-307. + 177. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + 271.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 749. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 994.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.13e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-1.07e3 + 620. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 - 433.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (492. + 853. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (75.6 - 131. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (781. - 451. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−13.90068620086077476535780569167, −12.59101745278538340471726594812, −11.66210220669844138847498938268, −10.77846970000065845609669488008, −9.949421041223564867846391238217, −8.385339309917734958738779624865, −6.62649403121557837913019592214, −4.79093987516211689430157602071, −3.59059816956902647537424703465, −0.847961582973718987215475535114,
2.46580445501006281520513080518, 5.35429269017135668678125789639, 6.38353789712902397976676888872, 7.45262435428018917463098140420, 8.381744071175914740044448502428, 10.64550184927911598451325091865, 11.49338990495853792617822853206, 12.47463125554911405195936197981, 13.91756521369825222292089558387, 15.15695919582856469819052237947