L(s) = 1 | + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (6 − 10.3i)5-s − 7.99·8-s + (−12 − 20.7i)10-s + (24 + 41.5i)11-s − 56·13-s + (−8 + 13.8i)16-s + (57 + 98.7i)17-s + (1 − 1.73i)19-s − 48·20-s + 96·22-s + (−60 + 103. i)23-s + (−9.5 − 16.4i)25-s + (−56 + 96.9i)26-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.536 − 0.929i)5-s − 0.353·8-s + (−0.379 − 0.657i)10-s + (0.657 + 1.13i)11-s − 1.19·13-s + (−0.125 + 0.216i)16-s + (0.813 + 1.40i)17-s + (0.0120 − 0.0209i)19-s − 0.536·20-s + 0.930·22-s + (−0.543 + 0.942i)23-s + (−0.0759 − 0.131i)25-s + (−0.422 + 0.731i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.148812516\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.148812516\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + 1.73i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-6 + 10.3i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-24 - 41.5i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 56T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-57 - 98.7i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-1 + 1.73i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (60 - 103. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 54T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-118 - 204. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (73 - 126. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 126T + 6.89e4T^{2} \) |
| 43 | \( 1 + 376T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-6 + 10.3i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-87 - 150. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (69 + 119. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-190 + 329. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-242 - 419. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 576T + 3.57e5T^{2} \) |
| 73 | \( 1 + (575 + 995. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (388 - 672. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 378T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-195 + 337. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.33e3T + 9.12e5T^{2} \) |
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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
−9.935701544149692363438345023187, −9.181115071686692938601913878219, −8.266613280009776784303157147613, −7.19254278864143682365348419750, −6.11346436661775963350640502755, −5.13158060715214338603530428372, −4.52646837043423810501434606529, −3.39515941283079181160923057581, −1.95990059510989396449734175574, −1.27736906756257935114135507374,
0.50340224632015936634852455496, 2.44698001843132376037635940762, 3.20636866839926964328854875056, 4.45486784181315420070080859124, 5.50607698969204585179748222972, 6.29173850874581406427328007487, 7.00780620700644231623354704790, 7.82263217262482659145075055596, 8.825543463473312514983365764538, 9.763792912353710620449209565304