L(s) = 1 | + 1.56·2-s + 0.629·3-s + 0.454·4-s − 2.16·5-s + 0.986·6-s − 5.08·7-s − 2.42·8-s − 2.60·9-s − 3.39·10-s − 11-s + 0.286·12-s − 2.97·13-s − 7.96·14-s − 1.36·15-s − 4.70·16-s + 17-s − 4.07·18-s − 6.04·19-s − 0.983·20-s − 3.20·21-s − 1.56·22-s + 2.54·23-s − 1.52·24-s − 0.314·25-s − 4.66·26-s − 3.52·27-s − 2.30·28-s + ⋯ |
L(s) = 1 | + 1.10·2-s + 0.363·3-s + 0.227·4-s − 0.967·5-s + 0.402·6-s − 1.92·7-s − 0.856·8-s − 0.867·9-s − 1.07·10-s − 0.301·11-s + 0.0826·12-s − 0.825·13-s − 2.12·14-s − 0.352·15-s − 1.17·16-s + 0.242·17-s − 0.961·18-s − 1.38·19-s − 0.219·20-s − 0.698·21-s − 0.334·22-s + 0.530·23-s − 0.311·24-s − 0.0629·25-s − 0.914·26-s − 0.679·27-s − 0.436·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.007282318335\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.007282318335\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
good | 2 | \( 1 - 1.56T + 2T^{2} \) |
| 3 | \( 1 - 0.629T + 3T^{2} \) |
| 5 | \( 1 + 2.16T + 5T^{2} \) |
| 7 | \( 1 + 5.08T + 7T^{2} \) |
| 13 | \( 1 + 2.97T + 13T^{2} \) |
| 19 | \( 1 + 6.04T + 19T^{2} \) |
| 23 | \( 1 - 2.54T + 23T^{2} \) |
| 29 | \( 1 - 1.74T + 29T^{2} \) |
| 31 | \( 1 + 4.76T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 1.39T + 41T^{2} \) |
| 47 | \( 1 + 3.18T + 47T^{2} \) |
| 53 | \( 1 + 6.48T + 53T^{2} \) |
| 59 | \( 1 + 2.15T + 59T^{2} \) |
| 61 | \( 1 + 3.33T + 61T^{2} \) |
| 67 | \( 1 - 5.09T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 - 9.72T + 79T^{2} \) |
| 83 | \( 1 + 3.17T + 83T^{2} \) |
| 89 | \( 1 + 1.09T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−7.82532030969240362332844832862, −6.73672606809225010548971580968, −6.58581477777018213830324490223, −5.57805550763280336266845442330, −5.06450044943962839802810144784, −3.97986553817714029929606161527, −3.62828815610555069574137966701, −2.96669611592272821381487939869, −2.37590733774123506368537323069, −0.03146452608456840005883355196,
0.03146452608456840005883355196, 2.37590733774123506368537323069, 2.96669611592272821381487939869, 3.62828815610555069574137966701, 3.97986553817714029929606161527, 5.06450044943962839802810144784, 5.57805550763280336266845442330, 6.58581477777018213830324490223, 6.73672606809225010548971580968, 7.82532030969240362332844832862