| L(s) = 1 | + (−1.91 − 0.578i)2-s + (−0.736 + 2.74i)3-s + (3.33 + 2.21i)4-s + (−4.54 − 2.08i)5-s + (2.99 − 4.83i)6-s + (−6.67 − 2.11i)7-s + (−5.09 − 6.16i)8-s + (0.780 + 0.450i)9-s + (7.49 + 6.62i)10-s + (11.1 − 6.45i)11-s + (−8.54 + 7.52i)12-s + (5.64 − 5.64i)13-s + (11.5 + 7.90i)14-s + (9.08 − 10.9i)15-s + (6.19 + 14.7i)16-s + (6.24 − 23.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.957 − 0.289i)2-s + (−0.245 + 0.916i)3-s + (0.832 + 0.553i)4-s + (−0.908 − 0.417i)5-s + (0.499 − 0.806i)6-s + (−0.953 − 0.301i)7-s + (−0.637 − 0.770i)8-s + (0.0867 + 0.0500i)9-s + (0.749 + 0.662i)10-s + (1.01 − 0.586i)11-s + (−0.711 + 0.627i)12-s + (0.434 − 0.434i)13-s + (0.825 + 0.564i)14-s + (0.605 − 0.730i)15-s + (0.387 + 0.922i)16-s + (0.367 − 1.37i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0905 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0905 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.361814 - 0.330401i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.361814 - 0.330401i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.91 + 0.578i)T \) |
| 5 | \( 1 + (4.54 + 2.08i)T \) |
| 7 | \( 1 + (6.67 + 2.11i)T \) |
| good | 3 | \( 1 + (0.736 - 2.74i)T + (-7.79 - 4.5i)T^{2} \) |
| 11 | \( 1 + (-11.1 + 6.45i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-5.64 + 5.64i)T - 169iT^{2} \) |
| 17 | \( 1 + (-6.24 + 23.3i)T + (-250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (17.6 + 10.2i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (4.49 + 16.7i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 + 39.2iT - 841T^{2} \) |
| 31 | \( 1 + (2.72 + 4.72i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-3.56 - 13.2i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + 8.11iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (9.25 - 9.25i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (18.4 + 68.7i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (26.3 - 98.1i)T + (-2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (30.1 - 17.3i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (71.8 + 41.4i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (0.606 - 2.26i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + 41.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-19.9 - 5.34i)T + (4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-40.0 + 69.3i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-12.3 - 12.3i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (-35.4 + 61.4i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (12.3 + 12.3i)T + 9.40e3iT^{2} \) |
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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
−12.31391811060654885898075684137, −11.41574690363932518365372505363, −10.56502184804966860194691283401, −9.560879760963223952719728160529, −8.789900454764598602284618720495, −7.52611697041588337116865040809, −6.31502958059451446628396041850, −4.35686625281020520669048218159, −3.29050953231454547463600565954, −0.47122429270065846851340929098,
1.59551271559872012207591470875, 3.69868054534445478854270239478, 6.23535310899345663232900704108, 6.71186659440224738223035103198, 7.74442341382377616478559975451, 8.854308351237247933647784021022, 9.994549171378477994762326292044, 11.12850680761574659272353556430, 12.18043928145132564287641762451, 12.70397525094989923323763836716
