| L(s) = 1 | + 1.41·2-s + 3-s + 0.585·5-s + 1.41·6-s − 0.585·7-s − 2.82·8-s + 9-s + 0.828·10-s − 0.414·11-s − 2.24·13-s − 0.828·14-s + 0.585·15-s − 4.00·16-s + 2.41·17-s + 1.41·18-s − 2.58·19-s − 0.585·21-s − 0.585·22-s + 1.41·23-s − 2.82·24-s − 4.65·25-s − 3.17·26-s + 27-s + 8.07·29-s + 0.828·30-s + ⋯ |
| L(s) = 1 | + 1.00·2-s + 0.577·3-s + 0.261·5-s + 0.577·6-s − 0.221·7-s − 0.999·8-s + 0.333·9-s + 0.261·10-s − 0.124·11-s − 0.621·13-s − 0.221·14-s + 0.151·15-s − 1.00·16-s + 0.585·17-s + 0.333·18-s − 0.593·19-s − 0.127·21-s − 0.124·22-s + 0.294·23-s − 0.577·24-s − 0.931·25-s − 0.621·26-s + 0.192·27-s + 1.49·29-s + 0.151·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.732504389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.732504389\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 5 | \( 1 - 0.585T + 5T^{2} \) |
| 7 | \( 1 + 0.585T + 7T^{2} \) |
| 11 | \( 1 + 0.414T + 11T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 - 2.41T + 17T^{2} \) |
| 19 | \( 1 + 2.58T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 8.07T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 7.48T + 37T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 - 7.58T + 47T^{2} \) |
| 53 | \( 1 - 1.17T + 53T^{2} \) |
| 59 | \( 1 - 8.48T + 59T^{2} \) |
| 61 | \( 1 - 6.65T + 61T^{2} \) |
| 67 | \( 1 + 6.48T + 67T^{2} \) |
| 71 | \( 1 + 4.07T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 3.65T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + 0.343T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−13.45695470008391059890228053777, −12.71163363372123667207009067189, −11.77294837495192627806218510543, −10.18278166973852136003053300517, −9.258714987889494237299891927528, −8.065991432819730803053101151210, −6.58052100325357270844172164728, −5.31085207599506180561686349009, −4.06319251959605651767610147604, −2.70227925347984232972683592642,
2.70227925347984232972683592642, 4.06319251959605651767610147604, 5.31085207599506180561686349009, 6.58052100325357270844172164728, 8.065991432819730803053101151210, 9.258714987889494237299891927528, 10.18278166973852136003053300517, 11.77294837495192627806218510543, 12.71163363372123667207009067189, 13.45695470008391059890228053777
