| L(s) = 1 | − 2.01·2-s + 2.07·4-s − 2.76·5-s − 0.611·7-s − 0.157·8-s + 5.58·10-s − 3.74·11-s + 3.41·13-s + 1.23·14-s − 3.83·16-s − 7.57·17-s + 0.763·19-s − 5.75·20-s + 7.56·22-s + 2.17·23-s + 2.66·25-s − 6.90·26-s − 1.27·28-s − 4.54·29-s + 8.18·31-s + 8.06·32-s + 15.2·34-s + 1.69·35-s − 4.22·37-s − 1.54·38-s + 0.435·40-s + 9.04·41-s + ⋯ |
| L(s) = 1 | − 1.42·2-s + 1.03·4-s − 1.23·5-s − 0.231·7-s − 0.0556·8-s + 1.76·10-s − 1.12·11-s + 0.948·13-s + 0.329·14-s − 0.959·16-s − 1.83·17-s + 0.175·19-s − 1.28·20-s + 1.61·22-s + 0.453·23-s + 0.532·25-s − 1.35·26-s − 0.240·28-s − 0.843·29-s + 1.47·31-s + 1.42·32-s + 2.62·34-s + 0.286·35-s − 0.694·37-s − 0.250·38-s + 0.0689·40-s + 1.41·41-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.01T + 2T^{2} \) |
| 5 | \( 1 + 2.76T + 5T^{2} \) |
| 7 | \( 1 + 0.611T + 7T^{2} \) |
| 11 | \( 1 + 3.74T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 + 7.57T + 17T^{2} \) |
| 19 | \( 1 - 0.763T + 19T^{2} \) |
| 23 | \( 1 - 2.17T + 23T^{2} \) |
| 29 | \( 1 + 4.54T + 29T^{2} \) |
| 31 | \( 1 - 8.18T + 31T^{2} \) |
| 37 | \( 1 + 4.22T + 37T^{2} \) |
| 41 | \( 1 - 9.04T + 41T^{2} \) |
| 43 | \( 1 - 1.75T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 5.60T + 53T^{2} \) |
| 59 | \( 1 - 1.51T + 59T^{2} \) |
| 61 | \( 1 - 3.02T + 61T^{2} \) |
| 67 | \( 1 + 7.65T + 67T^{2} \) |
| 71 | \( 1 + 6.78T + 71T^{2} \) |
| 73 | \( 1 - 0.572T + 73T^{2} \) |
| 79 | \( 1 - 9.21T + 79T^{2} \) |
| 83 | \( 1 + 6.18T + 83T^{2} \) |
| 89 | \( 1 + 4.60T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−7.81571614807429208585472248401, −7.23786458788515913146315155683, −6.62983317720914628978835834823, −5.68274680606902871429981855528, −4.55466850911811826553405010601, −4.07968865450269651109739805364, −2.96244484277558539540187457417, −2.13443700139784583025300891157, −0.849289687273241901211436224477, 0,
0.849289687273241901211436224477, 2.13443700139784583025300891157, 2.96244484277558539540187457417, 4.07968865450269651109739805364, 4.55466850911811826553405010601, 5.68274680606902871429981855528, 6.62983317720914628978835834823, 7.23786458788515913146315155683, 7.81571614807429208585472248401
