| L(s) = 1 | + (−0.252 + 0.495i)3-s + (3.68 + 3.38i)5-s + (−7.20 + 7.20i)7-s + (5.10 + 7.03i)9-s + (−4.56 − 3.31i)11-s + (−22.6 − 3.58i)13-s + (−2.60 + 0.969i)15-s + (−8.10 − 15.9i)17-s + (13.6 + 4.41i)19-s + (−1.75 − 5.39i)21-s + (−4.05 − 25.6i)23-s + (2.08 + 24.9i)25-s + (−9.72 + 1.53i)27-s + (−14.8 + 4.81i)29-s + (−10.7 + 33.0i)31-s + ⋯ |
| L(s) = 1 | + (−0.0842 + 0.165i)3-s + (0.736 + 0.676i)5-s + (−1.02 + 1.02i)7-s + (0.567 + 0.781i)9-s + (−0.415 − 0.301i)11-s + (−1.74 − 0.275i)13-s + (−0.173 + 0.0646i)15-s + (−0.476 − 0.936i)17-s + (0.715 + 0.232i)19-s + (−0.0834 − 0.256i)21-s + (−0.176 − 1.11i)23-s + (0.0834 + 0.996i)25-s + (−0.360 + 0.0570i)27-s + (−0.510 + 0.165i)29-s + (−0.346 + 1.06i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.348i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.152166 + 0.845223i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.152166 + 0.845223i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-3.68 - 3.38i)T \) |
| good | 3 | \( 1 + (0.252 - 0.495i)T + (-5.29 - 7.28i)T^{2} \) |
| 7 | \( 1 + (7.20 - 7.20i)T - 49iT^{2} \) |
| 11 | \( 1 + (4.56 + 3.31i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (22.6 + 3.58i)T + (160. + 52.2i)T^{2} \) |
| 17 | \( 1 + (8.10 + 15.9i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (-13.6 - 4.41i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + (4.05 + 25.6i)T + (-503. + 163. i)T^{2} \) |
| 29 | \( 1 + (14.8 - 4.81i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (10.7 - 33.0i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (-2.42 + 15.3i)T + (-1.30e3 - 423. i)T^{2} \) |
| 41 | \( 1 + (37.3 - 27.1i)T + (519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-31.8 - 31.8i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (8.16 + 4.16i)T + (1.29e3 + 1.78e3i)T^{2} \) |
| 53 | \( 1 + (12.8 - 25.2i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-23.3 - 32.0i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-11.5 - 8.37i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (24.8 + 48.7i)T + (-2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (-23.8 - 73.5i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-15.4 - 97.7i)T + (-5.06e3 + 1.64e3i)T^{2} \) |
| 79 | \( 1 + (110. - 35.8i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-13.6 + 6.95i)T + (4.04e3 - 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-43.0 + 59.2i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-67.9 - 34.6i)T + (5.53e3 + 7.61e3i)T^{2} \) |
| 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
−11.39599933799547242110228272232, −10.27454562767547810310538789473, −9.828132233683931082985441269718, −8.983593237489776915419158918957, −7.54895052144144141644735504187, −6.77954921115057271029250625612, −5.63752940857572013913217105555, −4.88803702908293131566388257745, −2.97549660960016450211002415075, −2.32657095002789773776251009247,
0.34764029252665868554434609784, 1.95833938815880512756278499377, 3.62229977073911369831425526187, 4.72018640552127241413465602891, 5.90763900002614817058516264709, 6.94960510244145051762577104616, 7.59514438685651208257181859935, 9.186306320362721576222225661530, 9.783650288367545791484097508584, 10.26730755783809444397148581409
