| L(s) = 1 | + 1.44·2-s − 1.76·3-s + 0.0914·4-s + 0.577·5-s − 2.54·6-s + 4.06·7-s − 2.76·8-s + 0.103·9-s + 0.835·10-s − 0.623·11-s − 0.161·12-s − 13-s + 5.88·14-s − 1.01·15-s − 4.17·16-s + 7.29·17-s + 0.149·18-s + 7.03·19-s + 0.0528·20-s − 7.16·21-s − 0.901·22-s − 0.557·23-s + 4.86·24-s − 4.66·25-s − 1.44·26-s + 5.10·27-s + 0.372·28-s + ⋯ |
| L(s) = 1 | + 1.02·2-s − 1.01·3-s + 0.0457·4-s + 0.258·5-s − 1.04·6-s + 1.53·7-s − 0.975·8-s + 0.0343·9-s + 0.264·10-s − 0.187·11-s − 0.0465·12-s − 0.277·13-s + 1.57·14-s − 0.262·15-s − 1.04·16-s + 1.76·17-s + 0.0351·18-s + 1.61·19-s + 0.0118·20-s − 1.56·21-s − 0.192·22-s − 0.116·23-s + 0.992·24-s − 0.933·25-s − 0.283·26-s + 0.982·27-s + 0.0703·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.028519834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.028519834\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 1.44T + 2T^{2} \) |
| 3 | \( 1 + 1.76T + 3T^{2} \) |
| 5 | \( 1 - 0.577T + 5T^{2} \) |
| 7 | \( 1 - 4.06T + 7T^{2} \) |
| 11 | \( 1 + 0.623T + 11T^{2} \) |
| 17 | \( 1 - 7.29T + 17T^{2} \) |
| 19 | \( 1 - 7.03T + 19T^{2} \) |
| 23 | \( 1 + 0.557T + 23T^{2} \) |
| 29 | \( 1 + 3.69T + 29T^{2} \) |
| 31 | \( 1 - 0.604T + 31T^{2} \) |
| 37 | \( 1 - 8.21T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 - 3.41T + 43T^{2} \) |
| 47 | \( 1 + 3.55T + 47T^{2} \) |
| 53 | \( 1 - 7.01T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 + 2.06T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 - 2.45T + 71T^{2} \) |
| 73 | \( 1 + 8.21T + 73T^{2} \) |
| 83 | \( 1 - 4.49T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−10.06004584232786384917764218581, −9.257967531520886721929348566624, −8.027592364918244414116141441220, −7.42810808374968129878095346757, −5.87447548577795420366641672616, −5.58577662270656480233394449445, −4.95269120828292134217571294834, −4.00684807512372318064272513968, −2.72711919178583848331131552258, −1.07025446172876331650538388549,
1.07025446172876331650538388549, 2.72711919178583848331131552258, 4.00684807512372318064272513968, 4.95269120828292134217571294834, 5.58577662270656480233394449445, 5.87447548577795420366641672616, 7.42810808374968129878095346757, 8.027592364918244414116141441220, 9.257967531520886721929348566624, 10.06004584232786384917764218581
