L(s) = 1 | + (1.22 − 0.707i)2-s + (1.81 + 2.38i)3-s + (0.999 − 1.73i)4-s + (−1.93 + 1.11i)5-s + (3.91 + 1.64i)6-s + (−2.08 + 6.68i)7-s − 2.82i·8-s + (−2.40 + 8.67i)9-s + (−1.58 + 2.73i)10-s + (14.2 + 8.25i)11-s + (5.95 − 0.757i)12-s + 13.0·13-s + (2.16 + 9.65i)14-s + (−6.18 − 2.59i)15-s + (−2.00 − 3.46i)16-s + (4.10 + 2.37i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.605 + 0.795i)3-s + (0.249 − 0.433i)4-s + (−0.387 + 0.223i)5-s + (0.652 + 0.273i)6-s + (−0.298 + 0.954i)7-s − 0.353i·8-s + (−0.267 + 0.963i)9-s + (−0.158 + 0.273i)10-s + (1.29 + 0.750i)11-s + (0.495 − 0.0631i)12-s + 1.00·13-s + (0.154 + 0.689i)14-s + (−0.412 − 0.172i)15-s + (−0.125 − 0.216i)16-s + (0.241 + 0.139i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.26189 + 0.966066i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.26189 + 0.966066i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.22 + 0.707i)T \) |
| 3 | \( 1 + (-1.81 - 2.38i)T \) |
| 5 | \( 1 + (1.93 - 1.11i)T \) |
| 7 | \( 1 + (2.08 - 6.68i)T \) |
good | 11 | \( 1 + (-14.2 - 8.25i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 13.0T + 169T^{2} \) |
| 17 | \( 1 + (-4.10 - 2.37i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (16.8 + 29.2i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (10.0 - 5.82i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 39.6iT - 841T^{2} \) |
| 31 | \( 1 + (-26.8 + 46.5i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (19.9 + 34.5i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 63.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 13.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-9.23 + 5.33i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-76.5 - 44.2i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (39.2 + 22.6i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-33.8 - 58.5i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (13.4 - 23.2i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 10.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-37.0 + 64.0i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (41.4 + 71.7i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 36.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (52.5 - 30.3i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 18.6T + 9.40e3T^{2} \) |
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
−12.19300987042373157966608520730, −11.35708781013583211517401251801, −10.42985302257600796320480855692, −9.228559566642534257465329610818, −8.691135040873223268997660418921, −7.04356898374239813592736678289, −5.83031208849247428645275935848, −4.45159761720252894313537967462, −3.56396276178189544134791668216, −2.22940839898687709035720274896,
1.25236657727735016472693498933, 3.41043729321187417026203805374, 4.10464243703888072692561323698, 6.16597562573225370600030338198, 6.68776671494131408719404773965, 8.045622131463465129545378016075, 8.557502316832561568595208629257, 10.03708530816394767009011001953, 11.43484821507108716103627379967, 12.19051608773783881518484361795