L(s) = 1 | + 4i·2-s + (15.5 − 0.805i)3-s − 16·4-s − 10.2·5-s + (3.22 + 62.2i)6-s − 139. i·7-s − 64i·8-s + (241. − 25.0i)9-s − 41.1i·10-s − 570.·11-s + (−249. + 12.8i)12-s − 754.·13-s + 559.·14-s + (−160. + 8.29i)15-s + 256·16-s − 143.·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.998 − 0.0516i)3-s − 0.5·4-s − 0.184·5-s + (0.0365 + 0.706i)6-s − 1.07i·7-s − 0.353i·8-s + (0.994 − 0.103i)9-s − 0.130i·10-s − 1.42·11-s + (−0.499 + 0.0258i)12-s − 1.23·13-s + 0.762·14-s + (−0.183 + 0.00951i)15-s + 0.250·16-s − 0.120·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.148 + 0.988i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.029170142\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.029170142\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 4iT \) |
| 3 | \( 1 + (-15.5 + 0.805i)T \) |
| 23 | \( 1 + (2.48e3 + 505. i)T \) |
good | 5 | \( 1 + 10.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 139. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 570.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 754.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 143.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.08e3iT - 2.47e6T^{2} \) |
| 29 | \( 1 - 5.65e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 2.62e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.24e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 965. iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.20e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.93e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.88e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.07e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 8.37e3iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 2.83e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 4.91e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 1.89e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.91e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 8.60e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.74e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.65e5iT - 8.58e9T^{2} \) |
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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
−12.34529639088110887460507131915, −10.57984534254310939727126186378, −9.829505004281361102727343543563, −8.582006219178130475520562918305, −7.53907432457988313238525439652, −7.06903218003908293632324053214, −5.12351375789673363883218962167, −3.99493648388046723156152401282, −2.46544037892390894852166970643, −0.28983447379749797072801867334,
2.05900458539464222128288479762, 2.84698383887825939534612653442, 4.35281858149395515829507226195, 5.69889701351384288953564936049, 7.74035700303390550534211920596, 8.294126749051194846293263515964, 9.689375181449762384508099610547, 10.14515802691537319963602818662, 11.75603619013500010756984073535, 12.49874691302882716711861165670