| L(s) = 1 | + (−3.54 + 3.84i)2-s + (−0.642 + 1.46i)3-s + (−1.59 − 19.3i)4-s + (−15.3 − 5.26i)5-s + (−3.36 − 7.66i)6-s + (−0.615 − 2.43i)7-s + (46.9 + 36.5i)8-s + (16.5 + 17.9i)9-s + (74.5 − 40.3i)10-s + (20.0 − 30.6i)11-s + (29.2 + 10.0i)12-s + (20.1 + 6.90i)13-s + (11.5 + 6.24i)14-s + (17.5 − 19.0i)15-s + (−154. + 25.7i)16-s + (−23.2 − 7.97i)17-s + ⋯ |
| L(s) = 1 | + (−1.25 + 1.36i)2-s + (−0.123 + 0.281i)3-s + (−0.199 − 2.41i)4-s + (−1.37 − 0.470i)5-s + (−0.228 − 0.521i)6-s + (−0.0332 − 0.131i)7-s + (2.07 + 1.61i)8-s + (0.613 + 0.666i)9-s + (2.35 − 1.27i)10-s + (0.548 − 0.839i)11-s + (0.704 + 0.241i)12-s + (0.429 + 0.147i)13-s + (0.220 + 0.119i)14-s + (0.302 − 0.328i)15-s + (−2.40 + 0.401i)16-s + (−0.331 − 0.113i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0798i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0148323 + 0.371070i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0148323 + 0.371070i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 229 | \( 1 + (2.50e3 - 2.39e3i)T \) |
| good | 2 | \( 1 + (3.54 - 3.84i)T + (-0.660 - 7.97i)T^{2} \) |
| 3 | \( 1 + (0.642 - 1.46i)T + (-18.2 - 19.8i)T^{2} \) |
| 5 | \( 1 + (15.3 + 5.26i)T + (98.6 + 76.7i)T^{2} \) |
| 7 | \( 1 + (0.615 + 2.43i)T + (-301. + 163. i)T^{2} \) |
| 11 | \( 1 + (-20.0 + 30.6i)T + (-534. - 1.21e3i)T^{2} \) |
| 13 | \( 1 + (-20.1 - 6.90i)T + (1.73e3 + 1.34e3i)T^{2} \) |
| 17 | \( 1 + (23.2 + 7.97i)T + (3.87e3 + 3.01e3i)T^{2} \) |
| 19 | \( 1 + (-39.1 + 13.4i)T + (5.41e3 - 4.21e3i)T^{2} \) |
| 23 | \( 1 + (115. + 62.5i)T + (6.65e3 + 1.01e4i)T^{2} \) |
| 29 | \( 1 + (-52.6 - 208. i)T + (-2.14e4 + 1.16e4i)T^{2} \) |
| 31 | \( 1 + (-78.5 + 120. i)T + (-1.19e4 - 2.72e4i)T^{2} \) |
| 37 | \( 1 + (185. + 30.9i)T + (4.79e4 + 1.64e4i)T^{2} \) |
| 41 | \( 1 + (267. - 290. i)T + (-5.69e3 - 6.86e4i)T^{2} \) |
| 43 | \( 1 + (123. + 20.6i)T + (7.51e4 + 2.58e4i)T^{2} \) |
| 47 | \( 1 + (-138. - 150. i)T + (-8.57e3 + 1.03e5i)T^{2} \) |
| 53 | \( 1 + (56.5 - 128. i)T + (-1.00e5 - 1.09e5i)T^{2} \) |
| 59 | \( 1 + (558. - 93.1i)T + (1.94e5 - 6.66e4i)T^{2} \) |
| 61 | \( 1 + (-538. - 584. i)T + (-1.87e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (-141. - 153. i)T + (-2.48e4 + 2.99e5i)T^{2} \) |
| 71 | \( 1 + (-610. - 934. i)T + (-1.43e5 + 3.27e5i)T^{2} \) |
| 73 | \( 1 + (247. + 192. i)T + (9.54e4 + 3.77e5i)T^{2} \) |
| 79 | \( 1 + (-49.9 + 197. i)T + (-4.33e5 - 2.34e5i)T^{2} \) |
| 83 | \( 1 + (797. - 133. i)T + (5.40e5 - 1.85e5i)T^{2} \) |
| 89 | \( 1 - 658.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (483. - 376. i)T + (2.24e5 - 8.84e5i)T^{2} \) |
| 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
−11.91633569880004772450882125250, −10.96710779075139876134936743508, −10.08747369917037068381976021940, −8.882584083570735373302722784506, −8.284702338963672415156557744935, −7.45138619449377534490576037079, −6.50399607847521314201330912995, −5.17193671716084162709170786903, −4.04265221908770013193889040461, −1.06942321509930212302204784720,
0.30310512964071132574861985781, 1.76484234470257031339283381030, 3.42340245102933552872442336230, 4.13308349689241077641361476532, 6.75286079971228139866869431015, 7.60688913670904985319155261339, 8.442615416151841735544654212020, 9.563096326671206352171861497012, 10.33088510201463543085313892606, 11.38870407722278836968018387114
