| L(s) = 1 | + 2.11·2-s + 2.47·4-s − 3.89·5-s − 3.33·7-s + 1.00·8-s − 8.23·10-s − 0.756·11-s + 2.48·13-s − 7.05·14-s − 2.82·16-s − 0.745·17-s − 4.48·19-s − 9.64·20-s − 1.60·22-s + 2.28·23-s + 10.1·25-s + 5.26·26-s − 8.25·28-s − 0.463·29-s − 3.94·31-s − 7.98·32-s − 1.57·34-s + 12.9·35-s − 4.75·37-s − 9.49·38-s − 3.92·40-s − 4.89·41-s + ⋯ |
| L(s) = 1 | + 1.49·2-s + 1.23·4-s − 1.74·5-s − 1.26·7-s + 0.356·8-s − 2.60·10-s − 0.228·11-s + 0.689·13-s − 1.88·14-s − 0.705·16-s − 0.180·17-s − 1.02·19-s − 2.15·20-s − 0.341·22-s + 0.476·23-s + 2.03·25-s + 1.03·26-s − 1.56·28-s − 0.0861·29-s − 0.708·31-s − 1.41·32-s − 0.270·34-s + 2.19·35-s − 0.781·37-s − 1.54·38-s − 0.620·40-s − 0.764·41-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
| \(L(1)\) |
\(\approx\) |
\(1.721136108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.721136108\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.11T + 2T^{2} \) |
| 5 | \( 1 + 3.89T + 5T^{2} \) |
| 7 | \( 1 + 3.33T + 7T^{2} \) |
| 11 | \( 1 + 0.756T + 11T^{2} \) |
| 13 | \( 1 - 2.48T + 13T^{2} \) |
| 17 | \( 1 + 0.745T + 17T^{2} \) |
| 19 | \( 1 + 4.48T + 19T^{2} \) |
| 23 | \( 1 - 2.28T + 23T^{2} \) |
| 29 | \( 1 + 0.463T + 29T^{2} \) |
| 31 | \( 1 + 3.94T + 31T^{2} \) |
| 37 | \( 1 + 4.75T + 37T^{2} \) |
| 41 | \( 1 + 4.89T + 41T^{2} \) |
| 43 | \( 1 - 5.94T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 - 4.14T + 59T^{2} \) |
| 61 | \( 1 - 0.267T + 61T^{2} \) |
| 67 | \( 1 + 1.91T + 67T^{2} \) |
| 71 | \( 1 + 6.75T + 71T^{2} \) |
| 73 | \( 1 - 7.93T + 73T^{2} \) |
| 79 | \( 1 - 4.31T + 79T^{2} \) |
| 83 | \( 1 - 2.54T + 83T^{2} \) |
| 89 | \( 1 - 5.38T + 89T^{2} \) |
| 97 | \( 1 - 9.48T + 97T^{2} \) |
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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
−7.73451747357971939363263351326, −7.05225589645685705699392673005, −6.55625005831177676402676370210, −5.81401789740620923539077255645, −4.99315508697583561609332208344, −4.13970210141085366603264074272, −3.74740370253773752547467205833, −3.22327456693995035381506926655, −2.34576838390782889459587935517, −0.50691068827284321942920618832,
0.50691068827284321942920618832, 2.34576838390782889459587935517, 3.22327456693995035381506926655, 3.74740370253773752547467205833, 4.13970210141085366603264074272, 4.99315508697583561609332208344, 5.81401789740620923539077255645, 6.55625005831177676402676370210, 7.05225589645685705699392673005, 7.73451747357971939363263351326
