| L(s) = 1 | + (2.24 − 1.78i)2-s + (−0.788 + 0.309i)3-s + (0.937 − 4.10i)4-s + (−0.381 − 0.0285i)5-s + (−1.21 + 2.10i)6-s + (−0.491 + 0.283i)7-s + (−0.264 − 0.548i)8-s + (−6.07 + 5.63i)9-s + (−0.905 + 0.617i)10-s + (−0.0636 − 0.278i)11-s + (0.531 + 3.52i)12-s + (−3.32 − 2.26i)13-s + (−0.594 + 1.51i)14-s + (0.309 − 0.0954i)15-s + (13.6 + 6.55i)16-s + (−0.557 − 7.44i)17-s + ⋯ |
| L(s) = 1 | + (1.12 − 0.893i)2-s + (−0.262 + 0.103i)3-s + (0.234 − 1.02i)4-s + (−0.0762 − 0.00571i)5-s + (−0.202 + 0.350i)6-s + (−0.0702 + 0.0405i)7-s + (−0.0330 − 0.0685i)8-s + (−0.674 + 0.625i)9-s + (−0.0905 + 0.0617i)10-s + (−0.00578 − 0.0253i)11-s + (0.0443 + 0.294i)12-s + (−0.255 − 0.174i)13-s + (−0.0424 + 0.108i)14-s + (0.0206 − 0.00636i)15-s + (0.850 + 0.409i)16-s + (−0.0328 − 0.437i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.44840 - 0.695791i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44840 - 0.695791i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (14.2 - 40.5i)T \) |
| good | 2 | \( 1 + (-2.24 + 1.78i)T + (0.890 - 3.89i)T^{2} \) |
| 3 | \( 1 + (0.788 - 0.309i)T + (6.59 - 6.12i)T^{2} \) |
| 5 | \( 1 + (0.381 + 0.0285i)T + (24.7 + 3.72i)T^{2} \) |
| 7 | \( 1 + (0.491 - 0.283i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (0.0636 + 0.278i)T + (-109. + 52.4i)T^{2} \) |
| 13 | \( 1 + (3.32 + 2.26i)T + (61.7 + 157. i)T^{2} \) |
| 17 | \( 1 + (0.557 + 7.44i)T + (-285. + 43.0i)T^{2} \) |
| 19 | \( 1 + (-12.5 + 13.4i)T + (-26.9 - 359. i)T^{2} \) |
| 23 | \( 1 + (15.7 + 4.84i)T + (437. + 297. i)T^{2} \) |
| 29 | \( 1 + (30.7 + 12.0i)T + (616. + 572. i)T^{2} \) |
| 31 | \( 1 + (-51.6 + 7.78i)T + (918. - 283. i)T^{2} \) |
| 37 | \( 1 + (-38.7 - 22.3i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (1.34 + 1.68i)T + (-374. + 1.63e3i)T^{2} \) |
| 47 | \( 1 + (-6.95 + 30.4i)T + (-1.99e3 - 958. i)T^{2} \) |
| 53 | \( 1 + (28.8 - 19.6i)T + (1.02e3 - 2.61e3i)T^{2} \) |
| 59 | \( 1 + (-67.7 - 32.6i)T + (2.17e3 + 2.72e3i)T^{2} \) |
| 61 | \( 1 + (12.7 - 84.8i)T + (-3.55e3 - 1.09e3i)T^{2} \) |
| 67 | \( 1 + (-6.38 - 5.92i)T + (335. + 4.47e3i)T^{2} \) |
| 71 | \( 1 + (15.4 + 50.0i)T + (-4.16e3 + 2.83e3i)T^{2} \) |
| 73 | \( 1 + (-26.3 + 38.6i)T + (-1.94e3 - 4.96e3i)T^{2} \) |
| 79 | \( 1 + (-60.1 - 104. i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (41.2 + 104. i)T + (-5.04e3 + 4.68e3i)T^{2} \) |
| 89 | \( 1 + (64.3 - 25.2i)T + (5.80e3 - 5.38e3i)T^{2} \) |
| 97 | \( 1 + (29.8 + 130. i)T + (-8.47e3 + 4.08e3i)T^{2} \) |
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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
−15.28457455153985723886232788648, −14.00854304194801527814579909371, −13.28910532912038651000583280838, −11.89593734219078044880939411237, −11.27182558178992495553175880354, −9.913665037894180368060592831832, −7.979113265019518929385835799950, −5.82144682673117765057988896264, −4.53234259967729648969031688838, −2.70677649751313320025932109387,
3.74718927315892657762428489366, 5.45927621799633964241431814019, 6.50039992326439559245636898723, 7.932136544890259845231828286369, 9.779464569259786012844145230777, 11.60440160257061677319457584426, 12.60159456417167263988460398650, 13.84237296068072317535999270632, 14.68213095274756209902394080461, 15.69478613315238550368690240444
