L(s) = 1 | + (−2 − 3.46i)2-s + (−7.99 + 13.8i)4-s + (33 − 57.1i)5-s + (−88 − 152. i)7-s + 63.9·8-s − 264·10-s + (30 + 51.9i)11-s + (329 − 569. i)13-s + (−352 + 609. i)14-s + (−128 − 221. i)16-s − 414·17-s + 956·19-s + (528 + 914. i)20-s + (120 − 207. i)22-s + (−300 + 519. i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.590 − 1.02i)5-s + (−0.678 − 1.17i)7-s + 0.353·8-s − 0.834·10-s + (0.0747 + 0.129i)11-s + (0.539 − 0.935i)13-s + (−0.479 + 0.831i)14-s + (−0.125 − 0.216i)16-s − 0.347·17-s + 0.607·19-s + (0.295 + 0.511i)20-s + (0.0528 − 0.0915i)22-s + (−0.118 + 0.204i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.029136369\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.029136369\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2 + 3.46i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-33 + 57.1i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (88 + 152. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-30 - 51.9i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-329 + 569. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + 414T + 1.41e6T^{2} \) |
| 19 | \( 1 - 956T + 2.47e6T^{2} \) |
| 23 | \( 1 + (300 - 519. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (2.78e3 + 4.82e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-1.79e3 + 3.11e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + 8.45e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (9.59e3 - 1.66e4i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (6.65e3 + 1.15e4i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-9.84e3 - 1.70e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + 3.12e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (1.31e4 - 2.28e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-1.55e4 - 2.69e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-8.40e3 + 1.45e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 6.12e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.55e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (3.72e4 + 6.44e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-3.23e3 - 5.60e3i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + 3.27e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (8.30e4 + 1.43e5i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−11.31083628981911357277529148944, −10.19435932028320191334937239994, −9.596607063018230719323874333645, −8.491477272410933951244570280360, −7.36752355199396529308542196344, −5.90548340631346778502403758501, −4.52055918032255361492384018215, −3.27578969823456919025205140140, −1.45705986404635375537244233502, −0.38569627695670245800613639958,
1.94604836494457412257745051942, 3.32801834741535461292429985093, 5.30190107945685754228367967023, 6.36609328641803433477892556839, 6.94912766991826614476950094662, 8.593277018327677442989062432588, 9.331592250788872040954456958300, 10.32023654798945410743985602112, 11.38348467459561190527222774911, 12.54590515680061547517802980223