L(s) = 1 | + 28.5·3-s + (5.81 + 55.5i)5-s − 92.3i·7-s + 571.·9-s + 541. i·11-s − 782.·13-s + (165. + 1.58e3i)15-s + 1.55e3i·17-s + 484. i·19-s − 2.63e3i·21-s + 1.09e3i·23-s + (−3.05e3 + 646. i)25-s + 9.37e3·27-s − 597. i·29-s + 8.86e3·31-s + ⋯ |
L(s) = 1 | + 1.83·3-s + (0.104 + 0.994i)5-s − 0.712i·7-s + 2.35·9-s + 1.34i·11-s − 1.28·13-s + (0.190 + 1.82i)15-s + 1.30i·17-s + 0.307i·19-s − 1.30i·21-s + 0.432i·23-s + (−0.978 + 0.206i)25-s + 2.47·27-s − 0.131i·29-s + 1.65·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(3.868007521\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.868007521\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-5.81 - 55.5i)T \) |
good | 3 | \( 1 - 28.5T + 243T^{2} \) |
| 7 | \( 1 + 92.3iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 541. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 782.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.55e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 484. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 1.09e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 597. iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 8.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.36e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.45e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.82e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.79e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.65e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.44e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 2.45e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 2.68e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.80e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.06e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 2.81e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.45e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.77e4iT - 8.58e9T^{2} \) |
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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
−10.48087329934147501662113949149, −10.04020679563920659306973999053, −9.260640486139737492611621426806, −7.896891584604558178360631185552, −7.48052975350074382776969066759, −6.59031229369734477846376409367, −4.54592954243622555910061006149, −3.65894436412172762479760330864, −2.56491594168193819895649786278, −1.73901012137946667133582897178,
0.76026987547521405744777247544, 2.29755122281758943691043730992, 2.99996433560818239636333103323, 4.37337966428956333919092329454, 5.41216263832617348484514786157, 7.04131427113581820709176418073, 8.087493647720426780673735328425, 8.762870939417416834774477109786, 9.279621990884185580029660221123, 10.16758518459506080664078274793