| L(s) = 1 | + 1.91·2-s + 1.65·4-s − 2.48·5-s − 4.55·7-s − 0.654·8-s − 4.75·10-s − 0.317·11-s − 3.40·13-s − 8.70·14-s − 4.56·16-s − 5.09·17-s − 1.24·19-s − 4.11·20-s − 0.607·22-s + 1.90·23-s + 1.17·25-s − 6.50·26-s − 7.54·28-s + 3.46·29-s + 6.78·31-s − 7.42·32-s − 9.75·34-s + 11.3·35-s + 8.53·37-s − 2.38·38-s + 1.62·40-s + 6.35·41-s + ⋯ |
| L(s) = 1 | + 1.35·2-s + 0.828·4-s − 1.11·5-s − 1.72·7-s − 0.231·8-s − 1.50·10-s − 0.0958·11-s − 0.943·13-s − 2.32·14-s − 1.14·16-s − 1.23·17-s − 0.285·19-s − 0.921·20-s − 0.129·22-s + 0.397·23-s + 0.234·25-s − 1.27·26-s − 1.42·28-s + 0.643·29-s + 1.21·31-s − 1.31·32-s − 1.67·34-s + 1.91·35-s + 1.40·37-s − 0.386·38-s + 0.256·40-s + 0.991·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.035906493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.035906493\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 1.91T + 2T^{2} \) |
| 5 | \( 1 + 2.48T + 5T^{2} \) |
| 7 | \( 1 + 4.55T + 7T^{2} \) |
| 11 | \( 1 + 0.317T + 11T^{2} \) |
| 13 | \( 1 + 3.40T + 13T^{2} \) |
| 17 | \( 1 + 5.09T + 17T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 23 | \( 1 - 1.90T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 - 6.78T + 31T^{2} \) |
| 37 | \( 1 - 8.53T + 37T^{2} \) |
| 41 | \( 1 - 6.35T + 41T^{2} \) |
| 43 | \( 1 + 6.09T + 43T^{2} \) |
| 47 | \( 1 + 4.07T + 47T^{2} \) |
| 53 | \( 1 + 7.76T + 53T^{2} \) |
| 59 | \( 1 + 3.96T + 59T^{2} \) |
| 61 | \( 1 - 2.45T + 61T^{2} \) |
| 67 | \( 1 + 3.44T + 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 + 4.22T + 73T^{2} \) |
| 79 | \( 1 + 7.24T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 + 3.73T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.79981066783109564202641467437, −6.99750570744605870506083196880, −6.45596365550995567675995270969, −5.99279577247799917429224014024, −4.79822821407932121619610886418, −4.45133753587221603733506091407, −3.68181289925839935117453997281, −2.97147521279875703657775875098, −2.46981461323015680775582001457, −0.39301732245341906210034059397,
0.39301732245341906210034059397, 2.46981461323015680775582001457, 2.97147521279875703657775875098, 3.68181289925839935117453997281, 4.45133753587221603733506091407, 4.79822821407932121619610886418, 5.99279577247799917429224014024, 6.45596365550995567675995270969, 6.99750570744605870506083196880, 7.79981066783109564202641467437
