L(s) = 1 | + (−0.366 − 1.36i)2-s + (−0.448 + 1.67i)3-s + (−1.73 + i)4-s + (−3.58 + 3.48i)5-s + 2.44·6-s + (3.00 − 6.32i)7-s + (2 + 1.99i)8-s + (−2.59 − 1.50i)9-s + (6.07 + 3.61i)10-s + (−1.95 − 3.38i)11-s + (−0.896 − 3.34i)12-s + (−8.93 − 8.93i)13-s + (−9.73 − 1.78i)14-s + (−4.22 − 7.55i)15-s + (1.99 − 3.46i)16-s + (−14.3 − 3.83i)17-s + ⋯ |
L(s) = 1 | + (−0.183 − 0.683i)2-s + (−0.149 + 0.557i)3-s + (−0.433 + 0.250i)4-s + (−0.716 + 0.697i)5-s + 0.408·6-s + (0.428 − 0.903i)7-s + (0.250 + 0.249i)8-s + (−0.288 − 0.166i)9-s + (0.607 + 0.361i)10-s + (−0.177 − 0.308i)11-s + (−0.0747 − 0.278i)12-s + (−0.687 − 0.687i)13-s + (−0.695 − 0.127i)14-s + (−0.281 − 0.503i)15-s + (0.124 − 0.216i)16-s + (−0.841 − 0.225i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.354i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.935 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0687204 - 0.375132i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0687204 - 0.375132i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.366 + 1.36i)T \) |
| 3 | \( 1 + (0.448 - 1.67i)T \) |
| 5 | \( 1 + (3.58 - 3.48i)T \) |
| 7 | \( 1 + (-3.00 + 6.32i)T \) |
good | 11 | \( 1 + (1.95 + 3.38i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (8.93 + 8.93i)T + 169iT^{2} \) |
| 17 | \( 1 + (14.3 + 3.83i)T + (250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (28.0 + 16.1i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-12.3 + 3.30i)T + (458. - 264.5i)T^{2} \) |
| 29 | \( 1 - 26.6iT - 841T^{2} \) |
| 31 | \( 1 + (17.0 + 29.4i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-1.67 - 6.25i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + 26.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-21.0 - 21.0i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-10.7 - 40.1i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-11.7 + 43.7i)T + (-2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (20.7 - 12.0i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-8.39 + 14.5i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-57.0 - 15.2i)T + (3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + 132.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-2.29 + 8.56i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-90.6 - 52.3i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-94.1 - 94.1i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (43.2 + 24.9i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-118. + 118. i)T - 9.40e3iT^{2} \) |
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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
−11.17973912144324171517432858062, −10.97278032396449102070392610958, −10.10278697064899716404068753606, −8.826931661099843284014421837139, −7.78222903365049813278475733094, −6.72400875978410821796827575191, −4.88393164668434567971635537877, −3.96874892064080638044854840745, −2.68049474369848652003316312835, −0.22289180705527761589741784040,
1.96601886466453838492462458119, 4.28985958509058440466537631266, 5.29946941481824969180251598375, 6.52040124007613772174017927591, 7.60442548019016149114033388795, 8.515324620974151126245364802251, 9.155858865637361888742869696791, 10.70318778852351185326902255761, 11.89424353991700065931902930642, 12.47048621708434667278376235620