| L(s) = 1 | + (0.238 + 0.891i)2-s + (0.613 − 2.93i)3-s + (2.72 − 1.57i)4-s + (−3.51 − 3.55i)5-s + (2.76 − 0.154i)6-s + (2.96 + 11.0i)7-s + (4.66 + 4.66i)8-s + (−8.24 − 3.60i)9-s + (2.32 − 3.98i)10-s + (−1.30 + 2.26i)11-s + (−2.94 − 8.97i)12-s + (−0.764 + 2.85i)13-s + (−9.15 + 5.28i)14-s + (−12.5 + 8.15i)15-s + (3.24 − 5.62i)16-s + (−13.6 + 13.6i)17-s + ⋯ |
| L(s) = 1 | + (0.119 + 0.445i)2-s + (0.204 − 0.978i)3-s + (0.681 − 0.393i)4-s + (−0.703 − 0.710i)5-s + (0.460 − 0.0257i)6-s + (0.423 + 1.57i)7-s + (0.583 + 0.583i)8-s + (−0.916 − 0.400i)9-s + (0.232 − 0.398i)10-s + (−0.118 + 0.205i)11-s + (−0.245 − 0.747i)12-s + (−0.0588 + 0.219i)13-s + (−0.653 + 0.377i)14-s + (−0.839 + 0.543i)15-s + (0.203 − 0.351i)16-s + (−0.805 + 0.805i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.26571 - 0.230162i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26571 - 0.230162i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.613 + 2.93i)T \) |
| 5 | \( 1 + (3.51 + 3.55i)T \) |
| good | 2 | \( 1 + (-0.238 - 0.891i)T + (-3.46 + 2i)T^{2} \) |
| 7 | \( 1 + (-2.96 - 11.0i)T + (-42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (1.30 - 2.26i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (0.764 - 2.85i)T + (-146. - 84.5i)T^{2} \) |
| 17 | \( 1 + (13.6 - 13.6i)T - 289iT^{2} \) |
| 19 | \( 1 + 11.4iT - 361T^{2} \) |
| 23 | \( 1 + (-2.89 + 10.7i)T + (-458. - 264.5i)T^{2} \) |
| 29 | \( 1 + (-23.0 - 13.2i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (21.8 + 37.8i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-14.4 + 14.4i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + (0.924 + 1.60i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (49.0 - 13.1i)T + (1.60e3 - 924.5i)T^{2} \) |
| 47 | \( 1 + (16.4 + 61.3i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-6.43 - 6.43i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-49.5 + 28.5i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-16.8 + 29.1i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-31.4 - 8.41i)T + (3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + 63.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + (50.9 + 50.9i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + (-59.5 - 34.4i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-101. + 27.2i)T + (5.96e3 - 3.44e3i)T^{2} \) |
| 89 | \( 1 - 136. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (9.45 + 35.3i)T + (-8.14e3 + 4.70e3i)T^{2} \) |
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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
−15.29589793299681507301363118091, −14.78311531567399495387712187843, −13.10271125156995230807901705839, −12.04707241582390719606420256457, −11.27561230996556476488450277180, −8.902322917078447611102380129796, −7.998114482945107417467024949376, −6.55986336429868575688899014653, −5.23171529269526181267104639689, −2.14341509506124769977558976845,
3.20756659148753286597031506137, 4.35555887948263220000064444778, 6.97866175666211000997873647180, 8.066419165716768498879865479030, 10.17705273012566847876875578078, 10.88702315451712184109797727092, 11.68706737946834592778777435960, 13.52150155695051627232906324223, 14.58092633965807724224020951439, 15.78773268062336861860231105880
