L(s) = 1 | + (4.5 + 2.59i)3-s + (8.5 − 16.4i)7-s + (13.5 + 23.3i)9-s + 91.7i·13-s + (109.5 − 63.2i)19-s + (81 − 51.9i)21-s + (62.5 − 108. i)25-s + 140. i·27-s + (163.5 + 94.3i)31-s + (161.5 + 279. i)37-s + (−238.5 + 413. i)39-s + 71·43-s + (−198.5 − 279. i)49-s + 657·57-s + (−810 + 467. i)61-s + ⋯ |
L(s) = 1 | + (0.866 + 0.499i)3-s + (0.458 − 0.888i)7-s + (0.5 + 0.866i)9-s + 1.95i·13-s + (1.32 − 0.763i)19-s + (0.841 − 0.539i)21-s + (0.5 − 0.866i)25-s + 1.00i·27-s + (0.947 + 0.546i)31-s + (0.717 + 1.24i)37-s + (−0.979 + 1.69i)39-s + 0.251·43-s + (−0.578 − 0.815i)49-s + 1.52·57-s + (−1.70 + 0.981i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.772041143\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.772041143\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.5 - 2.59i)T \) |
| 7 | \( 1 + (-8.5 + 16.4i)T \) |
good | 5 | \( 1 + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 91.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-109.5 + 63.2i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 2.43e4T^{2} \) |
| 31 | \( 1 + (-163.5 - 94.3i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-161.5 - 279. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 - 71T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (810 - 467. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (63.5 - 109. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 3.57e5T^{2} \) |
| 73 | \( 1 + (-1.05e3 - 608. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (693.5 + 1.20e3i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.37e3iT - 9.12e5T^{2} \) |
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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
−11.16746101750626670069516557682, −10.15644293581712758957229695857, −9.372328076866202656009549970451, −8.509991046590564321369141419154, −7.47380754882913727864428415743, −6.65433313606475283522592152497, −4.80677589142445616698285578104, −4.22052943577324992919404934687, −2.86830823917034559603212827896, −1.40197601495288475041682825195,
1.06303868262728222591978223276, 2.54891026651341968682770360274, 3.46926703289478087203484688916, 5.17658193791306964161874585166, 6.08387713149217653397674176283, 7.62543310731963777447804636875, 7.999309970429956483031113772457, 9.074455748968271670387811815193, 9.898436270261964888646303737860, 11.06537468872563047691192576599