L(s) = 1 | + (0.965 − 0.258i)2-s + (−0.258 − 0.965i)3-s + (0.866 − 0.499i)4-s + (−2.19 − 0.440i)5-s + (−0.499 − 0.866i)6-s + (−1.44 + 1.44i)7-s + (0.707 − 0.707i)8-s + (−0.866 + 0.499i)9-s + (−2.23 + 0.141i)10-s − 4.77·11-s + (−0.707 − 0.707i)12-s + (−6.13 − 1.64i)13-s + (−1.02 + 1.77i)14-s + (0.141 + 2.23i)15-s + (0.500 − 0.866i)16-s + (−0.624 − 2.33i)17-s + ⋯ |
L(s) = 1 | + (0.683 − 0.183i)2-s + (−0.149 − 0.557i)3-s + (0.433 − 0.249i)4-s + (−0.980 − 0.197i)5-s + (−0.204 − 0.353i)6-s + (−0.547 + 0.547i)7-s + (0.249 − 0.249i)8-s + (−0.288 + 0.166i)9-s + (−0.705 + 0.0448i)10-s − 1.44·11-s + (−0.204 − 0.204i)12-s + (−1.70 − 0.456i)13-s + (−0.273 + 0.474i)14-s + (0.0366 + 0.576i)15-s + (0.125 − 0.216i)16-s + (−0.151 − 0.565i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0333831 + 0.458678i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0333831 + 0.458678i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 3 | \( 1 + (0.258 + 0.965i)T \) |
| 5 | \( 1 + (2.19 + 0.440i)T \) |
| 19 | \( 1 + (-4.35 - 0.0647i)T \) |
good | 7 | \( 1 + (1.44 - 1.44i)T - 7iT^{2} \) |
| 11 | \( 1 + 4.77T + 11T^{2} \) |
| 13 | \( 1 + (6.13 + 1.64i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (0.624 + 2.33i)T + (-14.7 + 8.5i)T^{2} \) |
| 23 | \( 1 + (-0.619 + 2.31i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-0.138 - 0.240i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 1.20iT - 31T^{2} \) |
| 37 | \( 1 + (-2.38 - 2.38i)T + 37iT^{2} \) |
| 41 | \( 1 + (8.57 + 4.94i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.73 - 1.26i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (5.10 + 1.36i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.0540 - 0.0144i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-5.69 + 9.86i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.79 - 3.10i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.66 + 13.6i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-8.89 - 5.13i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.92 - 1.32i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-0.0784 + 0.135i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (9.57 + 9.57i)T + 83iT^{2} \) |
| 89 | \( 1 + (-3.92 - 6.80i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-15.3 + 4.11i)T + (84.0 - 48.5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.41428499512168733132915599627, −9.559695916535058143905866515609, −8.197774266031867390094422731013, −7.51881666129513722953713604911, −6.72974097280384615356982508790, −5.29316820418778598875480589507, −4.91773726866199869161970486376, −3.22954911345698847841791625401, −2.47882632559454743257440416796, −0.19492949327002167132650058511,
2.71303141251357909152475072698, 3.62913294723836633848546482190, 4.69098412922006777573401724785, 5.36876623571356122407232367122, 6.83039587131064472529803894474, 7.44758820357313154882260208695, 8.309225988479583316562719079944, 9.769302596555281715022511659775, 10.31716027465038411637783405030, 11.31824356420349932028203539794