| L(s) = 1 | + (1.54 + 1.27i)2-s + (1.09 − 1.33i)3-s + (0.762 + 3.92i)4-s + (0.121 + 0.400i)5-s + (3.39 − 0.667i)6-s + (0.354 − 1.78i)7-s + (−3.82 + 7.02i)8-s + (−0.585 − 2.94i)9-s + (−0.322 + 0.772i)10-s + (9.89 + 0.974i)11-s + (6.09 + 3.29i)12-s + (20.9 + 6.36i)13-s + (2.81 − 2.29i)14-s + (0.669 + 0.277i)15-s + (−14.8 + 5.98i)16-s + (2.44 + 5.90i)17-s + ⋯ |
| L(s) = 1 | + (0.771 + 0.636i)2-s + (0.366 − 0.446i)3-s + (0.190 + 0.981i)4-s + (0.0242 + 0.0800i)5-s + (0.566 − 0.111i)6-s + (0.0506 − 0.254i)7-s + (−0.477 + 0.878i)8-s + (−0.0650 − 0.326i)9-s + (−0.0322 + 0.0772i)10-s + (0.899 + 0.0886i)11-s + (0.507 + 0.274i)12-s + (1.61 + 0.489i)13-s + (0.201 − 0.164i)14-s + (0.0446 + 0.0184i)15-s + (−0.927 + 0.374i)16-s + (0.143 + 0.347i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.72523 + 1.51476i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.72523 + 1.51476i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.54 - 1.27i)T \) |
| 3 | \( 1 + (-1.09 + 1.33i)T \) |
| good | 5 | \( 1 + (-0.121 - 0.400i)T + (-20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (-0.354 + 1.78i)T + (-45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (-9.89 - 0.974i)T + (118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (-20.9 - 6.36i)T + (140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (-2.44 - 5.90i)T + (-204. + 204. i)T^{2} \) |
| 19 | \( 1 + (18.8 - 10.0i)T + (200. - 300. i)T^{2} \) |
| 23 | \( 1 + (-26.4 - 17.6i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (16.6 - 1.63i)T + (824. - 164. i)T^{2} \) |
| 31 | \( 1 + (5.41 - 5.41i)T - 961iT^{2} \) |
| 37 | \( 1 + (34.0 + 18.2i)T + (760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (8.53 + 5.70i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (14.2 + 17.4i)T + (-360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (33.8 + 81.6i)T + (-1.56e3 + 1.56e3i)T^{2} \) |
| 53 | \( 1 + (-45.4 - 4.48i)T + (2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (-23.0 - 76.0i)T + (-2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (-56.3 + 68.7i)T + (-725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (69.4 + 57.0i)T + (875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (88.3 + 17.5i)T + (4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (115. - 22.9i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-6.11 + 14.7i)T + (-4.41e3 - 4.41e3i)T^{2} \) |
| 83 | \( 1 + (3.35 + 6.28i)T + (-3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (-70.7 - 105. i)T + (-3.03e3 + 7.31e3i)T^{2} \) |
| 97 | \( 1 + (69.0 + 69.0i)T + 9.40e3iT^{2} \) |
| 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
−11.50005516017805873447844636972, −10.57580479529070388734446251914, −8.871824641762302017122246814423, −8.566499292165653416754299486447, −7.21284847105422974471396918087, −6.59198572730713138481396676403, −5.64612203310109399779139373370, −4.14560947333587424825306217956, −3.41461174414860441629377930606, −1.68146631996888490492909501263,
1.25781134803563714017842723399, 2.85865414319103143770787496705, 3.82113973346049526386600959475, 4.84874907089147681883066088046, 5.96278482283449207955159774870, 6.90752244280918450863118510958, 8.659297694482718133220881738009, 9.079330435286222789623017753232, 10.34722820428496430028042859335, 11.03034452702347345396195064986
