L(s) = 1 | + (4.5 + 2.59i)5-s + (3.17 + 5.49i)7-s + (8.17 − 4.71i)11-s + (−9.84 + 17.0i)13-s − 1.90i·17-s − 4.69·19-s + (8.17 + 4.71i)23-s + (1 + 1.73i)25-s + (2.84 − 1.64i)29-s + (−20.5 + 35.5i)31-s + 32.9i·35-s + 17.3·37-s + (53.5 + 30.9i)41-s + (0.477 + 0.826i)43-s + (−12.2 + 7.05i)47-s + ⋯ |
L(s) = 1 | + (0.900 + 0.519i)5-s + (0.453 + 0.785i)7-s + (0.743 − 0.429i)11-s + (−0.757 + 1.31i)13-s − 0.112i·17-s − 0.247·19-s + (0.355 + 0.205i)23-s + (0.0400 + 0.0692i)25-s + (0.0982 − 0.0567i)29-s + (−0.662 + 1.14i)31-s + 0.942i·35-s + 0.467·37-s + (1.30 + 0.754i)41-s + (0.0110 + 0.0192i)43-s + (−0.259 + 0.150i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.72157 + 1.09118i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.72157 + 1.09118i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-4.5 - 2.59i)T + (12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (-3.17 - 5.49i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-8.17 + 4.71i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (9.84 - 17.0i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 1.90iT - 289T^{2} \) |
| 19 | \( 1 + 4.69T + 361T^{2} \) |
| 23 | \( 1 + (-8.17 - 4.71i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-2.84 + 1.64i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (20.5 - 35.5i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 17.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-53.5 - 30.9i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-0.477 - 0.826i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (12.2 - 7.05i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 9.53iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-79.2 - 45.7i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-37.5 - 65.0i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-15.4 + 26.8i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 85.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 96.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-14.8 - 25.7i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (76.1 - 43.9i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 41.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (47.9 + 83.0i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.23697605863208314584637102324, −10.09909435099754860459584561233, −9.274109848955903287335680171356, −8.626913336618908173773457922061, −7.21839078051065151387208775637, −6.38455822614991526323218099808, −5.48923280818245499252296549719, −4.30828407233185125079706004098, −2.73341082197045377242081591206, −1.70241209274272067272022356836,
0.916666247310705112355900481396, 2.28633801740749565461564413357, 3.93986524255537947830211132552, 5.02259582802466201235199990554, 5.91035756359968016568483141297, 7.14042856532831997281312196859, 7.966537855803220177262467132523, 9.127335030323863647784042746214, 9.870828463184935676838240211020, 10.63979181261591131805922082710