| L(s) = 1 | + 9.34·2-s + 55.3·4-s − 69.4·5-s − 8.69·7-s + 218.·8-s − 649.·10-s + 970.·13-s − 81.2·14-s + 268.·16-s − 424.·17-s + 1.43e3·19-s − 3.84e3·20-s − 2.85e3·23-s + 1.69e3·25-s + 9.07e3·26-s − 481.·28-s − 7.46e3·29-s + 1.03e4·31-s − 4.47e3·32-s − 3.96e3·34-s + 603.·35-s + 167.·37-s + 1.33e4·38-s − 1.51e4·40-s + 5.68e3·41-s − 2.11e4·43-s − 2.66e4·46-s + ⋯ |
| L(s) = 1 | + 1.65·2-s + 1.72·4-s − 1.24·5-s − 0.0670·7-s + 1.20·8-s − 2.05·10-s + 1.59·13-s − 0.110·14-s + 0.261·16-s − 0.356·17-s + 0.909·19-s − 2.14·20-s − 1.12·23-s + 0.543·25-s + 2.63·26-s − 0.115·28-s − 1.64·29-s + 1.93·31-s − 0.772·32-s − 0.588·34-s + 0.0833·35-s + 0.0200·37-s + 1.50·38-s − 1.49·40-s + 0.527·41-s − 1.74·43-s − 1.85·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 9.34T + 32T^{2} \) |
| 5 | \( 1 + 69.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + 8.69T + 1.68e4T^{2} \) |
| 13 | \( 1 - 970.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 424.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.43e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.85e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.46e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.03e4T + 2.86e7T^{2} \) |
| 37 | \( 1 - 167.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.68e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.11e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.78e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.56e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.34e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.85e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.94e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.28e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.95e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.02e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.83e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.31e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.18e4T + 8.58e9T^{2} \) |
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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
−8.412219978484671576570460625703, −7.78325471365270315541151143760, −6.74265291539746109147672944841, −6.06571780414237712788348398343, −5.14999819710301333839233506918, −4.11401750158427420813884052159, −3.73321453603804647080292650254, −2.86610334391853253724189583284, −1.47241787667661047845335482400, 0,
1.47241787667661047845335482400, 2.86610334391853253724189583284, 3.73321453603804647080292650254, 4.11401750158427420813884052159, 5.14999819710301333839233506918, 6.06571780414237712788348398343, 6.74265291539746109147672944841, 7.78325471365270315541151143760, 8.412219978484671576570460625703
