L(s) = 1 | + (1.32 − 0.505i)2-s + (1.48 − 1.33i)4-s + (2.10 + 2.10i)5-s − 4.40·7-s + (1.28 − 2.51i)8-s + (3.84 + 1.71i)10-s + (−0.215 + 0.215i)11-s + (−2.73 − 2.73i)13-s + (−5.82 + 2.22i)14-s + (0.430 − 3.97i)16-s + 2.36i·17-s + (0.758 − 0.758i)19-s + (5.94 + 0.320i)20-s + (−0.175 + 0.393i)22-s − 1.75i·23-s + ⋯ |
L(s) = 1 | + (0.933 − 0.357i)2-s + (0.744 − 0.667i)4-s + (0.941 + 0.941i)5-s − 1.66·7-s + (0.456 − 0.889i)8-s + (1.21 + 0.542i)10-s + (−0.0650 + 0.0650i)11-s + (−0.758 − 0.758i)13-s + (−1.55 + 0.595i)14-s + (0.107 − 0.994i)16-s + 0.573i·17-s + (0.174 − 0.174i)19-s + (1.32 + 0.0717i)20-s + (−0.0374 + 0.0839i)22-s − 0.366i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.76228 - 0.294013i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.76228 - 0.294013i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.32 + 0.505i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-2.10 - 2.10i)T + 5iT^{2} \) |
| 7 | \( 1 + 4.40T + 7T^{2} \) |
| 11 | \( 1 + (0.215 - 0.215i)T - 11iT^{2} \) |
| 13 | \( 1 + (2.73 + 2.73i)T + 13iT^{2} \) |
| 17 | \( 1 - 2.36iT - 17T^{2} \) |
| 19 | \( 1 + (-0.758 + 0.758i)T - 19iT^{2} \) |
| 23 | \( 1 + 1.75iT - 23T^{2} \) |
| 29 | \( 1 + (5.54 - 5.54i)T - 29iT^{2} \) |
| 31 | \( 1 - 9.01iT - 31T^{2} \) |
| 37 | \( 1 + (-3.10 + 3.10i)T - 37iT^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + (3.54 + 3.54i)T + 43iT^{2} \) |
| 47 | \( 1 - 3.90T + 47T^{2} \) |
| 53 | \( 1 + (2.71 + 2.71i)T + 53iT^{2} \) |
| 59 | \( 1 + (-3.40 + 3.40i)T - 59iT^{2} \) |
| 61 | \( 1 + (1.75 + 1.75i)T + 61iT^{2} \) |
| 67 | \( 1 + (-9.11 + 9.11i)T - 67iT^{2} \) |
| 71 | \( 1 - 11.8iT - 71T^{2} \) |
| 73 | \( 1 + 0.482iT - 73T^{2} \) |
| 79 | \( 1 + 6.88iT - 79T^{2} \) |
| 83 | \( 1 + (-4.79 - 4.79i)T + 83iT^{2} \) |
| 89 | \( 1 + 7.00T + 89T^{2} \) |
| 97 | \( 1 + 3.34T + 97T^{2} \) |
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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
−12.87703332355453745395949373874, −12.53224179578012211433808275662, −10.86001923449454881357873558700, −10.20096827810122274040801516472, −9.421366104174466807929942604252, −7.15799245774610958890492758869, −6.38319020076084856130429720740, −5.42758558814453825204715243431, −3.48396164761634913952741917454, −2.52957698623826832091819826160,
2.51819270110812640098923984345, 4.15516711459903705594006560262, 5.54386122765817628547916916608, 6.34051154731749513323572370581, 7.56053618498693285674061338573, 9.273619415682010405566945382296, 9.778976656497763731254951888697, 11.51594547530035450589356959224, 12.55612919562958522833681767146, 13.21460542919134488713935173521