L(s) = 1 | + (0.866 − 0.5i)3-s + (−0.133 + 2.23i)5-s + (0.499 − 0.866i)9-s + (−2 − 3.46i)11-s − 2i·13-s + (1 + 1.99i)15-s + (1.73 − i)17-s + (−1 + 1.73i)19-s + (−5.19 − 3i)23-s + (−4.96 − 0.598i)25-s − 0.999i·27-s − 6·29-s + (−3 − 5.19i)31-s + (−3.46 − 1.99i)33-s + (3.46 + 2i)37-s + ⋯ |
L(s) = 1 | + (0.499 − 0.288i)3-s + (−0.0599 + 0.998i)5-s + (0.166 − 0.288i)9-s + (−0.603 − 1.04i)11-s − 0.554i·13-s + (0.258 + 0.516i)15-s + (0.420 − 0.242i)17-s + (−0.229 + 0.397i)19-s + (−1.08 − 0.625i)23-s + (−0.992 − 0.119i)25-s − 0.192i·27-s − 1.11·29-s + (−0.538 − 0.933i)31-s + (−0.603 − 0.348i)33-s + (0.569 + 0.328i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 + 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8881714720\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8881714720\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (0.133 - 2.23i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + (-1.73 + i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (3 + 5.19i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.46 - 2i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + (3.46 + 2i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.73 + i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (10.3 - 6i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + (12.1 - 7i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 + (-8 + 13.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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
−8.280012875117982510488016914379, −7.75195061424548306038158704563, −7.17214924081788713089749111376, −5.93626455951061175937823400600, −5.88050412939606147294929585546, −4.37426460407881051287396550506, −3.42338309065850821338364508709, −2.85520428056331316398722553554, −1.88350556712224397429343817048, −0.24171446108791185581727801293,
1.53500201938830136715055678266, 2.30600488266309567015034076772, 3.63416962705288610782299724849, 4.30850671565078949803722917557, 5.08295774549015950604946799226, 5.76547911886303417729779599806, 6.94384500647124698800297391187, 7.70973580323443419863822601132, 8.227998072614270052990858249509, 9.206042313400886797760568069374