| L(s) = 1 | + 1.81·2-s + 3.10·3-s + 1.28·4-s + 2.81·5-s + 5.62·6-s − 1.28·8-s + 6.62·9-s + 5.10·10-s − 3.10·11-s + 3.99·12-s + 8.72·15-s − 4.91·16-s + 0.524·17-s + 12.0·18-s + 0.813·19-s + 3.62·20-s − 5.62·22-s + 7.33·23-s − 4.00·24-s + 2.91·25-s + 11.2·27-s + 8.28·29-s + 15.8·30-s + 1.39·31-s − 6.33·32-s − 9.62·33-s + 0.951·34-s + ⋯ |
| L(s) = 1 | + 1.28·2-s + 1.79·3-s + 0.644·4-s + 1.25·5-s + 2.29·6-s − 0.455·8-s + 2.20·9-s + 1.61·10-s − 0.935·11-s + 1.15·12-s + 2.25·15-s − 1.22·16-s + 0.127·17-s + 2.83·18-s + 0.186·19-s + 0.811·20-s − 1.19·22-s + 1.53·23-s − 0.816·24-s + 0.583·25-s + 2.16·27-s + 1.53·29-s + 2.89·30-s + 0.250·31-s − 1.12·32-s − 1.67·33-s + 0.163·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.893048943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.893048943\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.81T + 2T^{2} \) |
| 3 | \( 1 - 3.10T + 3T^{2} \) |
| 5 | \( 1 - 2.81T + 5T^{2} \) |
| 11 | \( 1 + 3.10T + 11T^{2} \) |
| 17 | \( 1 - 0.524T + 17T^{2} \) |
| 19 | \( 1 - 0.813T + 19T^{2} \) |
| 23 | \( 1 - 7.33T + 23T^{2} \) |
| 29 | \( 1 - 8.28T + 29T^{2} \) |
| 31 | \( 1 - 1.39T + 31T^{2} \) |
| 37 | \( 1 - 6.15T + 37T^{2} \) |
| 41 | \( 1 + 4.20T + 41T^{2} \) |
| 43 | \( 1 - 6.75T + 43T^{2} \) |
| 47 | \( 1 + 5.97T + 47T^{2} \) |
| 53 | \( 1 + 2.49T + 53T^{2} \) |
| 59 | \( 1 + 4.47T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 - 8.72T + 71T^{2} \) |
| 73 | \( 1 + 2.34T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 1.18T + 97T^{2} \) |
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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
−7.85995459832320819837846184204, −6.99513608344172690540322173047, −6.38491433425356465756221500644, −5.53942925144540527765526344285, −4.87253910406839158401923624218, −4.28990263982267676262659249156, −3.24412042499830330446531350310, −2.80935165752218388585585009821, −2.34412177363160750335198883222, −1.29412056216131083000370105970,
1.29412056216131083000370105970, 2.34412177363160750335198883222, 2.80935165752218388585585009821, 3.24412042499830330446531350310, 4.28990263982267676262659249156, 4.87253910406839158401923624218, 5.53942925144540527765526344285, 6.38491433425356465756221500644, 6.99513608344172690540322173047, 7.85995459832320819837846184204
