| L(s) = 1 | + 9.92·2-s + 66.4·4-s + 47.3·5-s − 2.03·7-s + 342.·8-s + 469.·10-s − 183.·11-s + 729.·13-s − 20.1·14-s + 1.26e3·16-s − 1.21e3·17-s − 473.·19-s + 3.14e3·20-s − 1.82e3·22-s + 3.61e3·23-s − 886.·25-s + 7.23e3·26-s − 134.·28-s + 1.32e3·29-s − 5.18e3·31-s + 1.62e3·32-s − 1.20e4·34-s − 96.0·35-s − 1.47e4·37-s − 4.69e3·38-s + 1.61e4·40-s − 6.31e3·41-s + ⋯ |
| L(s) = 1 | + 1.75·2-s + 2.07·4-s + 0.846·5-s − 0.0156·7-s + 1.88·8-s + 1.48·10-s − 0.457·11-s + 1.19·13-s − 0.0274·14-s + 1.23·16-s − 1.01·17-s − 0.300·19-s + 1.75·20-s − 0.803·22-s + 1.42·23-s − 0.283·25-s + 2.09·26-s − 0.0325·28-s + 0.292·29-s − 0.968·31-s + 0.281·32-s − 1.78·34-s − 0.0132·35-s − 1.76·37-s − 0.527·38-s + 1.59·40-s − 0.587·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.362485672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.362485672\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 9.92T + 32T^{2} \) |
| 5 | \( 1 - 47.3T + 3.12e3T^{2} \) |
| 7 | \( 1 + 2.03T + 1.68e4T^{2} \) |
| 11 | \( 1 + 183.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 729.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.21e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 473.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.61e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.18e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.47e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.31e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.13e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.16e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.22e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.95e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.03e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.34e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 123.T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.52e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.81e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.25e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.86e4T + 8.58e9T^{2} \) |
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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
−13.35418281511745110465052884949, −12.81310639765973884250458484306, −11.40417746819265014992767181103, −10.55718670715410987686197140764, −8.844681727226755833252423371855, −6.93982598025648920719091652752, −5.95069203559678384691961081154, −4.90503067824908720958695030086, −3.43828729503495798370368934469, −1.97498695601199816331444717692,
1.97498695601199816331444717692, 3.43828729503495798370368934469, 4.90503067824908720958695030086, 5.95069203559678384691961081154, 6.93982598025648920719091652752, 8.844681727226755833252423371855, 10.55718670715410987686197140764, 11.40417746819265014992767181103, 12.81310639765973884250458484306, 13.35418281511745110465052884949
