L(s) = 1 | + (0.0494 + 5.65i)2-s − 10.7i·3-s + (−31.9 + 0.559i)4-s − 25i·5-s + (60.7 − 0.531i)6-s + 198.·7-s + (−4.74 − 180. i)8-s + 127.·9-s + (141. − 1.23i)10-s − 85.9i·11-s + (6.01 + 343. i)12-s − 407. i·13-s + (9.83 + 1.12e3i)14-s − 268.·15-s + (1.02e3 − 35.8i)16-s + 1.20e3·17-s + ⋯ |
L(s) = 1 | + (0.00874 + 0.999i)2-s − 0.689i·3-s + (−0.999 + 0.0174i)4-s − 0.447i·5-s + (0.689 − 0.00602i)6-s + 1.53·7-s + (−0.0262 − 0.999i)8-s + 0.524·9-s + (0.447 − 0.00391i)10-s − 0.214i·11-s + (0.0120 + 0.689i)12-s − 0.668i·13-s + (0.0134 + 1.53i)14-s − 0.308·15-s + (0.999 − 0.0349i)16-s + 1.01·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0262i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.68967 + 0.0221702i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.68967 + 0.0221702i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0494 - 5.65i)T \) |
| 5 | \( 1 + 25iT \) |
good | 3 | \( 1 + 10.7iT - 243T^{2} \) |
| 7 | \( 1 - 198.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 85.9iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 407. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.20e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 206. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 2.59e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.19e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 1.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.47e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.80e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.26e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.43e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.27e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.07e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 1.13e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 6.26e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 6.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.32e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.91e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.80e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 3.01e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.13e5T + 8.58e9T^{2} \) |
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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
−15.09076834302304086982255081775, −14.08762265228919770765945838604, −13.03291455163885037720912771417, −11.84293611097104954255846776616, −9.969692525099070805420964102474, −8.189689846030477180178868862217, −7.68522175019223183184109891378, −5.89017948863040940061067559255, −4.50160930087904594174855467603, −1.18163940505072871177166965518,
1.75341189498998976147199956875, 3.90744714601063730703799695672, 5.12268857001657364394650960783, 7.72665860462029115285702781507, 9.244164907051766853331155633014, 10.45308157249826425327899799114, 11.28941013375291602322210236281, 12.47562789300324280504342583796, 14.19022828430160957722805490122, 14.70147901918569284439048223952