| L(s) = 1 | + 1.30·2-s − 1.30·3-s − 0.302·4-s + 3.60·5-s − 1.69·6-s − 3·8-s − 1.30·9-s + 4.69·10-s + 0.394·12-s − 0.605·13-s − 4.69·15-s − 3.30·16-s + 6.30·17-s − 1.69·18-s − 3·19-s − 1.09·20-s − 6.30·23-s + 3.90·24-s + 7.99·25-s − 0.788·26-s + 5.60·27-s − 8.30·29-s − 6.11·30-s − 31-s + 1.69·32-s + 8.21·34-s + 0.394·36-s + ⋯ |
| L(s) = 1 | + 0.921·2-s − 0.752·3-s − 0.151·4-s + 1.61·5-s − 0.692·6-s − 1.06·8-s − 0.434·9-s + 1.48·10-s + 0.113·12-s − 0.167·13-s − 1.21·15-s − 0.825·16-s + 1.52·17-s − 0.400·18-s − 0.688·19-s − 0.244·20-s − 1.31·23-s + 0.797·24-s + 1.59·25-s − 0.154·26-s + 1.07·27-s − 1.54·29-s − 1.11·30-s − 0.179·31-s + 0.300·32-s + 1.40·34-s + 0.0657·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.30T + 2T^{2} \) |
| 3 | \( 1 + 1.30T + 3T^{2} \) |
| 5 | \( 1 - 3.60T + 5T^{2} \) |
| 13 | \( 1 + 0.605T + 13T^{2} \) |
| 17 | \( 1 - 6.30T + 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 + 6.30T + 23T^{2} \) |
| 29 | \( 1 + 8.30T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 - 9.21T + 37T^{2} \) |
| 41 | \( 1 - 7T + 41T^{2} \) |
| 43 | \( 1 + 5.30T + 43T^{2} \) |
| 47 | \( 1 + 8.90T + 47T^{2} \) |
| 53 | \( 1 + 2.09T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + 0.697T + 61T^{2} \) |
| 67 | \( 1 + 9.51T + 67T^{2} \) |
| 71 | \( 1 + 0.697T + 71T^{2} \) |
| 73 | \( 1 + 5T + 73T^{2} \) |
| 79 | \( 1 - 4.69T + 79T^{2} \) |
| 83 | \( 1 - 3T + 83T^{2} \) |
| 89 | \( 1 + 17.7T + 89T^{2} \) |
| 97 | \( 1 + 3.60T + 97T^{2} \) |
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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
−7.67350169139566866917842929542, −6.48229684571363421016591261122, −5.94692700100053647366424764717, −5.71805591498903709958178308395, −5.06174233819472935436022293022, −4.26694779860717531370681081431, −3.26204536787864239143267894272, −2.47330961011786419743308123253, −1.44955327214707115234210094964, 0,
1.44955327214707115234210094964, 2.47330961011786419743308123253, 3.26204536787864239143267894272, 4.26694779860717531370681081431, 5.06174233819472935436022293022, 5.71805591498903709958178308395, 5.94692700100053647366424764717, 6.48229684571363421016591261122, 7.67350169139566866917842929542
