L(s) = 1 | + (1.22 − 0.707i)2-s + (0.999 − 1.73i)4-s + (−3.82 + 2.20i)5-s + (−2.62 − 0.358i)7-s − 2.82i·8-s + (−3.12 + 5.40i)10-s + (−5.04 − 2.91i)11-s + (−3.46 + 1.41i)14-s + (−2.00 − 3.46i)16-s + 8.82i·20-s − 8.24·22-s + (7.24 − 12.5i)25-s + (−3.24 + 4.18i)28-s + 7.58i·29-s + (−0.378 + 0.655i)31-s + (−4.89 − 2.82i)32-s + ⋯ |
L(s) = 1 | + (0.866 − 0.499i)2-s + (0.499 − 0.866i)4-s + (−1.70 + 0.987i)5-s + (−0.990 − 0.135i)7-s − 0.999i·8-s + (−0.987 + 1.70i)10-s + (−1.52 − 0.878i)11-s + (−0.925 + 0.377i)14-s + (−0.500 − 0.866i)16-s + 1.97i·20-s − 1.75·22-s + (1.44 − 2.50i)25-s + (−0.612 + 0.790i)28-s + 1.40i·29-s + (−0.0680 + 0.117i)31-s + (−0.866 − 0.499i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0332449 + 0.332382i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0332449 + 0.332382i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.22 + 0.707i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.62 + 0.358i)T \) |
good | 5 | \( 1 + (3.82 - 2.20i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (5.04 + 2.91i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 7.58iT - 29T^{2} \) |
| 31 | \( 1 + (0.378 - 0.655i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.52 + 2.03i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.89 + 1.67i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.86 + 15.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12.1iT - 83T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 17.9T + 97T^{2} \) |
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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
−10.77094226278805963851364815379, −10.05117957638941729038414960136, −8.532862801159932028188235329679, −7.45966158337519045211612504908, −6.78940226118516075849565598044, −5.67864350040100179506955387195, −4.39983156338771042715490629319, −3.27141121622401510867074093909, −2.93362571980390290213624692251, −0.14016251005132513698023917893,
2.74774382084406092953052371020, 3.89972900788846604767602282237, 4.67948044863502671991038598619, 5.59909080930352825590496823391, 6.96148212405879362967414927338, 7.74372541652705191014184922232, 8.295032046750335848700924674853, 9.468984867324106230187210781376, 10.75958835362566212617400322169, 11.76226125872886005558344169998