| L(s) = 1 | + 2.31·2-s − 0.203·3-s + 3.36·4-s − 0.470·6-s − 0.683·7-s + 3.16·8-s − 2.95·9-s − 3.68·11-s − 0.683·12-s + 4.43·13-s − 1.58·14-s + 0.593·16-s − 6.85·18-s − 1.03·19-s + 0.138·21-s − 8.52·22-s + 4.52·23-s − 0.642·24-s + 10.2·26-s + 1.21·27-s − 2.30·28-s + 3.69·29-s + 10.8·31-s − 4.94·32-s + 0.747·33-s − 9.95·36-s − 0.308·37-s + ⋯ |
| L(s) = 1 | + 1.63·2-s − 0.117·3-s + 1.68·4-s − 0.192·6-s − 0.258·7-s + 1.11·8-s − 0.986·9-s − 1.10·11-s − 0.197·12-s + 1.23·13-s − 0.423·14-s + 0.148·16-s − 1.61·18-s − 0.238·19-s + 0.0303·21-s − 1.81·22-s + 0.943·23-s − 0.131·24-s + 2.01·26-s + 0.233·27-s − 0.434·28-s + 0.685·29-s + 1.95·31-s − 0.874·32-s + 0.130·33-s − 1.65·36-s − 0.0507·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.612531638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.612531638\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 2.31T + 2T^{2} \) |
| 3 | \( 1 + 0.203T + 3T^{2} \) |
| 7 | \( 1 + 0.683T + 7T^{2} \) |
| 11 | \( 1 + 3.68T + 11T^{2} \) |
| 13 | \( 1 - 4.43T + 13T^{2} \) |
| 19 | \( 1 + 1.03T + 19T^{2} \) |
| 23 | \( 1 - 4.52T + 23T^{2} \) |
| 29 | \( 1 - 3.69T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 + 0.308T + 37T^{2} \) |
| 41 | \( 1 - 6.15T + 41T^{2} \) |
| 43 | \( 1 - 7.88T + 43T^{2} \) |
| 47 | \( 1 - 4.43T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 + 9.94T + 61T^{2} \) |
| 67 | \( 1 - 9.16T + 67T^{2} \) |
| 71 | \( 1 - 9.37T + 71T^{2} \) |
| 73 | \( 1 - 2.26T + 73T^{2} \) |
| 79 | \( 1 + 7.42T + 79T^{2} \) |
| 83 | \( 1 + 8.92T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−7.80131461213482543915398581272, −6.87492425542217107303676584689, −6.16696732755652405373433842794, −5.82388131717622492796094326779, −5.05231030649148509776954017822, −4.45517441294456658423568827493, −3.57400430273574232342913175873, −2.85834359714150818727395242821, −2.40926676008057815465134965185, −0.835143812074610718485028871422,
0.835143812074610718485028871422, 2.40926676008057815465134965185, 2.85834359714150818727395242821, 3.57400430273574232342913175873, 4.45517441294456658423568827493, 5.05231030649148509776954017822, 5.82388131717622492796094326779, 6.16696732755652405373433842794, 6.87492425542217107303676584689, 7.80131461213482543915398581272
