| L(s) = 1 | + 1.34·2-s − 0.184·4-s + 1.65·5-s + 2.41·7-s − 2.94·8-s + 2.22·10-s + 5.94·11-s − 3.22·13-s + 3.24·14-s − 3.59·16-s + 3·17-s − 6.63·19-s − 0.305·20-s + 8.00·22-s − 2.94·23-s − 2.26·25-s − 4.34·26-s − 0.445·28-s − 1.29·29-s − 0.588·31-s + 1.04·32-s + 4.04·34-s + 3.98·35-s + 0.0418·37-s − 8.94·38-s − 4.86·40-s − 4.90·41-s + ⋯ |
| L(s) = 1 | + 0.952·2-s − 0.0923·4-s + 0.739·5-s + 0.911·7-s − 1.04·8-s + 0.704·10-s + 1.79·11-s − 0.894·13-s + 0.868·14-s − 0.899·16-s + 0.727·17-s − 1.52·19-s − 0.0682·20-s + 1.70·22-s − 0.613·23-s − 0.453·25-s − 0.852·26-s − 0.0842·28-s − 0.239·29-s − 0.105·31-s + 0.184·32-s + 0.693·34-s + 0.673·35-s + 0.00688·37-s − 1.45·38-s − 0.769·40-s − 0.765·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.038764129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.038764129\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.34T + 2T^{2} \) |
| 5 | \( 1 - 1.65T + 5T^{2} \) |
| 7 | \( 1 - 2.41T + 7T^{2} \) |
| 11 | \( 1 - 5.94T + 11T^{2} \) |
| 13 | \( 1 + 3.22T + 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 6.63T + 19T^{2} \) |
| 23 | \( 1 + 2.94T + 23T^{2} \) |
| 29 | \( 1 + 1.29T + 29T^{2} \) |
| 31 | \( 1 + 0.588T + 31T^{2} \) |
| 37 | \( 1 - 0.0418T + 37T^{2} \) |
| 41 | \( 1 + 4.90T + 41T^{2} \) |
| 43 | \( 1 + 5.18T + 43T^{2} \) |
| 47 | \( 1 + 3.73T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + 7.34T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 1.85T + 67T^{2} \) |
| 71 | \( 1 + 5.51T + 71T^{2} \) |
| 73 | \( 1 - 5.55T + 73T^{2} \) |
| 79 | \( 1 + 3.78T + 79T^{2} \) |
| 83 | \( 1 - 3.98T + 83T^{2} \) |
| 89 | \( 1 - 8.15T + 89T^{2} \) |
| 97 | \( 1 - 0.260T + 97T^{2} \) |
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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
−12.12434283161197637132816183329, −11.58577676527337891910551330172, −10.13096140643082686485371177664, −9.250945483838276770216079088236, −8.289762168637513029507742100679, −6.73434674165298570847267662000, −5.80728829618544993629238015979, −4.73753752546484056772824916487, −3.77960377445360964289304377066, −1.95715542249369011083465327052,
1.95715542249369011083465327052, 3.77960377445360964289304377066, 4.73753752546484056772824916487, 5.80728829618544993629238015979, 6.73434674165298570847267662000, 8.289762168637513029507742100679, 9.250945483838276770216079088236, 10.13096140643082686485371177664, 11.58577676527337891910551330172, 12.12434283161197637132816183329
