| L(s) = 1 | + 0.0298·2-s − 3-s − 1.99·4-s + 3.06·5-s − 0.0298·6-s − 1.69·7-s − 0.119·8-s + 9-s + 0.0912·10-s + 4.60·11-s + 1.99·12-s − 13-s − 0.0504·14-s − 3.06·15-s + 3.99·16-s + 3.22·17-s + 0.0298·18-s + 3.92·19-s − 6.11·20-s + 1.69·21-s + 0.137·22-s + 4.33·23-s + 0.119·24-s + 4.36·25-s − 0.0298·26-s − 27-s + 3.38·28-s + ⋯ |
| L(s) = 1 | + 0.0210·2-s − 0.577·3-s − 0.999·4-s + 1.36·5-s − 0.0121·6-s − 0.639·7-s − 0.0421·8-s + 0.333·9-s + 0.0288·10-s + 1.38·11-s + 0.577·12-s − 0.277·13-s − 0.0134·14-s − 0.790·15-s + 0.998·16-s + 0.782·17-s + 0.00702·18-s + 0.901·19-s − 1.36·20-s + 0.369·21-s + 0.0292·22-s + 0.903·23-s + 0.0243·24-s + 0.872·25-s − 0.00584·26-s − 0.192·27-s + 0.639·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.704181198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.704181198\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0298T + 2T^{2} \) |
| 5 | \( 1 - 3.06T + 5T^{2} \) |
| 7 | \( 1 + 1.69T + 7T^{2} \) |
| 11 | \( 1 - 4.60T + 11T^{2} \) |
| 17 | \( 1 - 3.22T + 17T^{2} \) |
| 19 | \( 1 - 3.92T + 19T^{2} \) |
| 23 | \( 1 - 4.33T + 23T^{2} \) |
| 29 | \( 1 + 0.592T + 29T^{2} \) |
| 31 | \( 1 + 4.96T + 31T^{2} \) |
| 37 | \( 1 + 2.80T + 37T^{2} \) |
| 41 | \( 1 + 3.44T + 41T^{2} \) |
| 43 | \( 1 - 8.82T + 43T^{2} \) |
| 47 | \( 1 + 0.175T + 47T^{2} \) |
| 53 | \( 1 + 0.255T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 5.34T + 67T^{2} \) |
| 71 | \( 1 - 7.61T + 71T^{2} \) |
| 73 | \( 1 - 2.64T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 + 4.88T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 - 8.65T + 97T^{2} \) |
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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
−8.786851402275737866645635946206, −7.60573721020906188278129102970, −6.79930240445387809980668757691, −6.09078650452686739230101382166, −5.49129014433911342543272416787, −4.90942958939410875186086564257, −3.85651477455297169554341043666, −3.11874921527200484481940193055, −1.69274668632087987457944626995, −0.830322573932325299601984249077,
0.830322573932325299601984249077, 1.69274668632087987457944626995, 3.11874921527200484481940193055, 3.85651477455297169554341043666, 4.90942958939410875186086564257, 5.49129014433911342543272416787, 6.09078650452686739230101382166, 6.79930240445387809980668757691, 7.60573721020906188278129102970, 8.786851402275737866645635946206
