| L(s) = 1 | + 2.69·2-s − 0.699·3-s + 5.24·4-s − 1.88·6-s − 4.85·7-s + 8.74·8-s − 2.51·9-s + 2.43·11-s − 3.67·12-s + 1.74·13-s − 13.0·14-s + 13.0·16-s − 6.76·18-s + 0.0462·19-s + 3.39·21-s + 6.54·22-s + 2.68·23-s − 6.11·24-s + 4.71·26-s + 3.85·27-s − 25.4·28-s − 1.97·29-s + 2.91·31-s + 17.6·32-s − 1.70·33-s − 13.1·36-s + 3.44·37-s + ⋯ |
| L(s) = 1 | + 1.90·2-s − 0.403·3-s + 2.62·4-s − 0.768·6-s − 1.83·7-s + 3.09·8-s − 0.836·9-s + 0.733·11-s − 1.05·12-s + 0.485·13-s − 3.49·14-s + 3.26·16-s − 1.59·18-s + 0.0106·19-s + 0.741·21-s + 1.39·22-s + 0.560·23-s − 1.24·24-s + 0.923·26-s + 0.741·27-s − 4.81·28-s − 0.367·29-s + 0.523·31-s + 3.12·32-s − 0.296·33-s − 2.19·36-s + 0.566·37-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
| \(L(1)\) |
\(\approx\) |
\(5.286132864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.286132864\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.69T + 2T^{2} \) |
| 3 | \( 1 + 0.699T + 3T^{2} \) |
| 7 | \( 1 + 4.85T + 7T^{2} \) |
| 11 | \( 1 - 2.43T + 11T^{2} \) |
| 13 | \( 1 - 1.74T + 13T^{2} \) |
| 19 | \( 1 - 0.0462T + 19T^{2} \) |
| 23 | \( 1 - 2.68T + 23T^{2} \) |
| 29 | \( 1 + 1.97T + 29T^{2} \) |
| 31 | \( 1 - 2.91T + 31T^{2} \) |
| 37 | \( 1 - 3.44T + 37T^{2} \) |
| 41 | \( 1 + 5.32T + 41T^{2} \) |
| 43 | \( 1 - 7.98T + 43T^{2} \) |
| 47 | \( 1 - 9.75T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 - 7.13T + 59T^{2} \) |
| 61 | \( 1 - 8.70T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 + 4.48T + 71T^{2} \) |
| 73 | \( 1 - 7.71T + 73T^{2} \) |
| 79 | \( 1 + 3.97T + 79T^{2} \) |
| 83 | \( 1 + 0.548T + 83T^{2} \) |
| 89 | \( 1 + 0.475T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−7.36086771785428223509636720870, −6.84162812308423307969275318899, −6.22291146330679644266009619846, −5.86225020938118144115014846611, −5.23997389421635348831927488415, −4.16360311732626904270522270941, −3.69966251749363725798011038234, −2.96822493112704676211774353706, −2.39545663311251914667263970223, −0.875053636361439058937259334265,
0.875053636361439058937259334265, 2.39545663311251914667263970223, 2.96822493112704676211774353706, 3.69966251749363725798011038234, 4.16360311732626904270522270941, 5.23997389421635348831927488415, 5.86225020938118144115014846611, 6.22291146330679644266009619846, 6.84162812308423307969275318899, 7.36086771785428223509636720870
