| L(s) = 1 | + (−2 + 3.46i)2-s + (4.84 − 8.38i)3-s + (−7.99 − 13.8i)4-s + (−23.1 + 40.0i)5-s + (19.3 + 33.5i)6-s + 177.·7-s + 63.9·8-s + (74.5 + 129. i)9-s + (−92.4 − 160. i)10-s + 5.50·11-s − 155.·12-s + (515. + 893. i)13-s + (−355. + 615. i)14-s + (223. + 387. i)15-s + (−128 + 221. i)16-s + (−466. + 808. i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.310 − 0.538i)3-s + (−0.249 − 0.433i)4-s + (−0.413 + 0.716i)5-s + (0.219 + 0.380i)6-s + 1.37·7-s + 0.353·8-s + (0.306 + 0.531i)9-s + (−0.292 − 0.506i)10-s + 0.0137·11-s − 0.310·12-s + (0.846 + 1.46i)13-s + (−0.484 + 0.839i)14-s + (0.257 + 0.445i)15-s + (−0.125 + 0.216i)16-s + (−0.391 + 0.678i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.32317 + 0.771753i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32317 + 0.771753i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2 - 3.46i)T \) |
| 19 | \( 1 + (-1.04e3 + 1.17e3i)T \) |
| good | 3 | \( 1 + (-4.84 + 8.38i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (23.1 - 40.0i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 - 177.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 5.50T + 1.61e5T^{2} \) |
| 13 | \( 1 + (-515. - 893. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + (466. - 808. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 23 | \( 1 + (1.77e3 + 3.06e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (2.08e3 + 3.61e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 - 3.76e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.92e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-2.49e3 + 4.32e3i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (7.27e3 - 1.25e4i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (7.67e3 + 1.33e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-1.02e4 - 1.78e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (6.11e3 - 1.05e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (9.09e3 + 1.57e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.13e4 + 1.97e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (3.37e4 - 5.85e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (-4.04e4 + 7.00e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-4.33e4 + 7.50e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 2.94e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (2.30e4 + 3.99e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (2.22e3 - 3.86e3i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
| 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
−15.39414418915818549228605644409, −14.33774371407446620314086470814, −13.56531905046109231538570203937, −11.63455272315715130825946503241, −10.66198439876017860795357983786, −8.721627304324880676529064176486, −7.72835669233260264172632464784, −6.60820292002172376565884811213, −4.51888841996129625372003946333, −1.78627131878609163724350701519,
1.15190701828481075698500113392, 3.62612698012495748851960256023, 5.09431534339023913786118802487, 7.85329927973001900647758806383, 8.746186640836378373550261357035, 10.12197499563529782304556832105, 11.38647067089225544454121673828, 12.42960661225906817916138703143, 13.86155222937751720545583428512, 15.23918457245311561683187100805
