| L(s) = 1 | + 3.46·2-s − 500.·4-s + 1.40e3·5-s + 1.66e3·7-s − 3.50e3·8-s + 4.86e3·10-s − 6.36e4·11-s − 1.11e5·13-s + 5.76e3·14-s + 2.43e5·16-s − 3.83e5·17-s + 7.44e3·19-s − 7.02e5·20-s − 2.20e5·22-s − 2.61e6·23-s + 1.81e4·25-s − 3.85e5·26-s − 8.32e5·28-s + 5.79e5·29-s + 3.51e6·31-s + 2.63e6·32-s − 1.32e6·34-s + 2.33e6·35-s − 9.82e6·37-s + 2.57e4·38-s − 4.92e6·40-s + 1.77e7·41-s + ⋯ |
| L(s) = 1 | + 0.153·2-s − 0.976·4-s + 1.00·5-s + 0.262·7-s − 0.302·8-s + 0.153·10-s − 1.31·11-s − 1.07·13-s + 0.0401·14-s + 0.930·16-s − 1.11·17-s + 0.0131·19-s − 0.981·20-s − 0.200·22-s − 1.95·23-s + 0.00927·25-s − 0.165·26-s − 0.256·28-s + 0.152·29-s + 0.684·31-s + 0.444·32-s − 0.170·34-s + 0.263·35-s − 0.862·37-s + 0.00200·38-s − 0.303·40-s + 0.982·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 3.46T + 512T^{2} \) |
| 5 | \( 1 - 1.40e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.66e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 6.36e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.11e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.83e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 7.44e3T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.61e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.79e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.51e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 9.82e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.77e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.91e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.67e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.04e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.34e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.40e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 7.31e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.72e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.64e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.44e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.73e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.69e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 5.86e8T + 7.60e17T^{2} \) |
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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
−14.30234790729116116484513664120, −13.51067767934819817156012728782, −12.40971542639178122594534419477, −10.41861738197732362496582320144, −9.424988568944930459500046386393, −7.932532008646692753027514193156, −5.82148488940544645794540919867, −4.58906370197315135486009635355, −2.31410530135178468844760205666, 0,
2.31410530135178468844760205666, 4.58906370197315135486009635355, 5.82148488940544645794540919867, 7.932532008646692753027514193156, 9.424988568944930459500046386393, 10.41861738197732362496582320144, 12.40971542639178122594534419477, 13.51067767934819817156012728782, 14.30234790729116116484513664120
