L(s) = 1 | + (1.22 − 1.22i)3-s + (−1.77 + 4.67i)5-s + (−2.55 − 2.55i)7-s − 2.99i·9-s + 8.24·11-s + (12.2 − 12.2i)13-s + (3.55 + 7.89i)15-s + (−12.4 − 12.4i)17-s + 34.4i·19-s − 6.24·21-s + (17.3 − 17.3i)23-s + (−18.6 − 16.5i)25-s + (−3.67 − 3.67i)27-s + 9.75i·29-s − 28.4·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (−0.355 + 0.934i)5-s + (−0.364 − 0.364i)7-s − 0.333i·9-s + 0.749·11-s + (0.942 − 0.942i)13-s + (0.236 + 0.526i)15-s + (−0.732 − 0.732i)17-s + 1.81i·19-s − 0.297·21-s + (0.754 − 0.754i)23-s + (−0.747 − 0.663i)25-s + (−0.136 − 0.136i)27-s + 0.336i·29-s − 0.919·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 + 0.560i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.828 + 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.037004600\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.037004600\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 + 1.22i)T \) |
| 5 | \( 1 + (1.77 - 4.67i)T \) |
good | 7 | \( 1 + (2.55 + 2.55i)T + 49iT^{2} \) |
| 11 | \( 1 - 8.24T + 121T^{2} \) |
| 13 | \( 1 + (-12.2 + 12.2i)T - 169iT^{2} \) |
| 17 | \( 1 + (12.4 + 12.4i)T + 289iT^{2} \) |
| 19 | \( 1 - 34.4iT - 361T^{2} \) |
| 23 | \( 1 + (-17.3 + 17.3i)T - 529iT^{2} \) |
| 29 | \( 1 - 9.75iT - 841T^{2} \) |
| 31 | \( 1 + 28.4T + 961T^{2} \) |
| 37 | \( 1 + (-7.34 - 7.34i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 74.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-34.8 + 34.8i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-22.0 - 22.0i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-64.6 + 64.6i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + 15.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 53.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-4.69 - 4.69i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 117.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-34.1 + 34.1i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 0.494iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (18.3 - 18.3i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 136. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-94.5 - 94.5i)T + 9.40e3iT^{2} \) |
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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
−9.784530741077700670407336211840, −8.826837477970858439636730568466, −8.006331986318653927657543683275, −7.18027988167753327467120492321, −6.51720309630259524572219723909, −5.65603812479539593999166943991, −3.98919757821594056177408267138, −3.45160011374008280916204155613, −2.30682172970505307251739476681, −0.77461638591474504386589210609,
1.06341336262662115359415274069, 2.44516821256787993865059307735, 3.86883323569346725598487834273, 4.34663683592873409050074528758, 5.49875545696013409434976480351, 6.49348676935980863901362980351, 7.44476297320848940354961057474, 8.590862424834922209681584575187, 9.238612058530551257052486981386, 9.308671507616102510919434371407