| L(s) = 1 | + 1.73·2-s + 0.999·4-s + 3.46·5-s − 7-s − 1.73·8-s + 5.99·10-s − 3.46·11-s + 5·13-s − 1.73·14-s − 5·16-s − 19-s + 3.46·20-s − 5.99·22-s − 6.92·23-s + 6.99·25-s + 8.66·26-s − 0.999·28-s − 3.46·29-s + 5·31-s − 5.19·32-s − 3.46·35-s − 37-s − 1.73·38-s − 6.00·40-s + 3.46·41-s − 43-s − 3.46·44-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.499·4-s + 1.54·5-s − 0.377·7-s − 0.612·8-s + 1.89·10-s − 1.04·11-s + 1.38·13-s − 0.462·14-s − 1.25·16-s − 0.229·19-s + 0.774·20-s − 1.27·22-s − 1.44·23-s + 1.39·25-s + 1.69·26-s − 0.188·28-s − 0.643·29-s + 0.898·31-s − 0.918·32-s − 0.585·35-s − 0.164·37-s − 0.280·38-s − 0.948·40-s + 0.541·41-s − 0.152·43-s − 0.522·44-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
| \(L(1)\) |
\(\approx\) |
\(2.376216278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.376216278\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.73T + 2T^{2} \) |
| 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 - 6.92T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 17T + 97T^{2} \) |
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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
−12.60055799547426009652810274468, −11.27363637433942366154952747032, −10.21723916378989491163277649923, −9.388168687038182801682601425010, −8.223590322904053967862384369796, −6.38988047348155697162740997975, −5.94367436151325889775645239570, −4.95763268889965373286462212722, −3.52168957571220695631214088920, −2.21095194874431803639976263612,
2.21095194874431803639976263612, 3.52168957571220695631214088920, 4.95763268889965373286462212722, 5.94367436151325889775645239570, 6.38988047348155697162740997975, 8.223590322904053967862384369796, 9.388168687038182801682601425010, 10.21723916378989491163277649923, 11.27363637433942366154952747032, 12.60055799547426009652810274468
