L(s) = 1 | + (5.94 − 5.35i)2-s − 40.3·3-s + (6.65 − 63.6i)4-s + (−32.9 + 120. i)5-s + (−239. + 216. i)6-s − 450.·7-s + (−301. − 413. i)8-s + 900.·9-s + (450. + 893. i)10-s − 390. i·11-s + (−268. + 2.56e3i)12-s − 3.23e3i·13-s + (−2.67e3 + 2.41e3i)14-s + (1.32e3 − 4.86e3i)15-s + (−4.00e3 − 846. i)16-s + 4.93e3i·17-s + ⋯ |
L(s) = 1 | + (0.742 − 0.669i)2-s − 1.49·3-s + (0.103 − 0.994i)4-s + (−0.263 + 0.964i)5-s + (−1.11 + 1.00i)6-s − 1.31·7-s + (−0.588 − 0.808i)8-s + 1.23·9-s + (0.450 + 0.893i)10-s − 0.293i·11-s + (−0.155 + 1.48i)12-s − 1.47i·13-s + (−0.976 + 0.879i)14-s + (0.393 − 1.44i)15-s + (−0.978 − 0.206i)16-s + 1.00i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.362i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.932 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0490070 + 0.261378i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0490070 + 0.261378i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-5.94 + 5.35i)T \) |
| 5 | \( 1 + (32.9 - 120. i)T \) |
good | 3 | \( 1 + 40.3T + 729T^{2} \) |
| 7 | \( 1 + 450.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 390. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 3.23e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 4.93e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 1.98e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 1.07e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + 2.11e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 3.32e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 1.41e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 4.28e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 5.77e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.38e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + 2.77e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 2.81e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 3.40e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 2.30e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 2.14e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 2.11e4iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 3.65e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 5.02e4T + 3.26e11T^{2} \) |
| 89 | \( 1 + 1.77e5T + 4.96e11T^{2} \) |
| 97 | \( 1 - 1.15e6iT - 8.32e11T^{2} \) |
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
−16.11570163748568795171899486683, −15.06999734713814582096350644999, −13.18530951354070879188285418929, −12.19777035513854529012551183714, −10.86725078577511623810838554896, −10.18238082679883836245026637005, −6.65823293350356333376255759705, −5.64867474895637494755770009725, −3.38145940820094893436617863347, −0.15441053523788744307457679807,
4.34537756149248488421851805698, 5.78190056324382634594970508093, 7.02783608005270510680527636835, 9.354025172854356479855238473867, 11.61140141749927054871918038482, 12.37609295931810388758255100739, 13.52699709556226811721005967162, 15.64678391612605360936745519763, 16.56792471664494044511631940873, 16.93723586081355791612218562290