L(s) = 1 | + (1.70 + 0.329i)3-s + (−1.87 − 1.08i)5-s − 3.41·7-s + (2.78 + 1.12i)9-s + 2.39i·11-s + (−4.61 + 2.66i)13-s + (−2.82 − 2.45i)15-s + (−5.15 − 2.97i)17-s + (−1.09 + 4.21i)19-s + (−5.80 − 1.12i)21-s + (3.01 − 1.74i)23-s + (−0.159 − 0.276i)25-s + (4.36 + 2.82i)27-s + (−4.68 − 8.10i)29-s + 4.72i·31-s + ⋯ |
L(s) = 1 | + (0.981 + 0.190i)3-s + (−0.837 − 0.483i)5-s − 1.28·7-s + (0.927 + 0.373i)9-s + 0.723i·11-s + (−1.27 + 0.738i)13-s + (−0.730 − 0.634i)15-s + (−1.25 − 0.722i)17-s + (−0.251 + 0.967i)19-s + (−1.26 − 0.245i)21-s + (0.628 − 0.362i)23-s + (−0.0319 − 0.0553i)25-s + (0.839 + 0.543i)27-s + (−0.869 − 1.50i)29-s + 0.848i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.340i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0617497 + 0.351892i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0617497 + 0.351892i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.70 - 0.329i)T \) |
| 19 | \( 1 + (1.09 - 4.21i)T \) |
good | 5 | \( 1 + (1.87 + 1.08i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 - 2.39iT - 11T^{2} \) |
| 13 | \( 1 + (4.61 - 2.66i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (5.15 + 2.97i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-3.01 + 1.74i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.68 + 8.10i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.72iT - 31T^{2} \) |
| 37 | \( 1 - 1.54iT - 37T^{2} \) |
| 41 | \( 1 + (4.63 - 8.02i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.248 - 0.430i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (7.18 - 4.15i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.46 - 7.73i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.192 + 0.332i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.38 + 2.40i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.99 + 2.88i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.344 + 0.596i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.06 + 7.04i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.09 - 3.52i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 1.73iT - 83T^{2} \) |
| 89 | \( 1 + (8.86 + 15.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (12.3 + 7.12i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.00783336764960712348771153201, −9.642174064631952864608402902089, −8.896536279595313963434128243531, −7.965934065284298473558917699622, −7.17593864818348759390509143307, −6.48219102105331798390209325292, −4.74236474767169109341406488677, −4.25537083653609059927661733450, −3.13332502221646017691628121753, −2.10946858842979253119024688645,
0.13687804326411518100094298571, 2.36798381753694205562189713145, 3.27318919910669394575386499167, 3.88025989767480188298248285218, 5.28530786982012359780621021359, 6.78024424379786452869149749072, 7.02776930611182928761001110193, 8.050718144986247669333811669435, 8.907143063784552710819631235760, 9.554288229440903335891651772589