| L(s) = 1 | + 1.65·3-s − 1.83·5-s + 2.08·7-s − 0.257·9-s + 0.951·11-s − 3.58·13-s − 3.04·15-s − 6.16·17-s + 7.24·19-s + 3.44·21-s − 3.21·23-s − 1.62·25-s − 5.39·27-s + 9.38·29-s − 3.00·31-s + 1.57·33-s − 3.82·35-s + 5.86·37-s − 5.93·39-s + 3.25·41-s + 0.290·43-s + 0.471·45-s − 3.79·47-s − 2.66·49-s − 10.2·51-s + 4.88·53-s − 1.74·55-s + ⋯ |
| L(s) = 1 | + 0.956·3-s − 0.821·5-s + 0.786·7-s − 0.0856·9-s + 0.286·11-s − 0.994·13-s − 0.785·15-s − 1.49·17-s + 1.66·19-s + 0.752·21-s − 0.671·23-s − 0.325·25-s − 1.03·27-s + 1.74·29-s − 0.539·31-s + 0.274·33-s − 0.645·35-s + 0.964·37-s − 0.950·39-s + 0.507·41-s + 0.0442·43-s + 0.0703·45-s − 0.554·47-s − 0.381·49-s − 1.43·51-s + 0.671·53-s − 0.235·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 1.65T + 3T^{2} \) |
| 5 | \( 1 + 1.83T + 5T^{2} \) |
| 7 | \( 1 - 2.08T + 7T^{2} \) |
| 11 | \( 1 - 0.951T + 11T^{2} \) |
| 13 | \( 1 + 3.58T + 13T^{2} \) |
| 17 | \( 1 + 6.16T + 17T^{2} \) |
| 19 | \( 1 - 7.24T + 19T^{2} \) |
| 23 | \( 1 + 3.21T + 23T^{2} \) |
| 29 | \( 1 - 9.38T + 29T^{2} \) |
| 31 | \( 1 + 3.00T + 31T^{2} \) |
| 37 | \( 1 - 5.86T + 37T^{2} \) |
| 41 | \( 1 - 3.25T + 41T^{2} \) |
| 43 | \( 1 - 0.290T + 43T^{2} \) |
| 47 | \( 1 + 3.79T + 47T^{2} \) |
| 53 | \( 1 - 4.88T + 53T^{2} \) |
| 59 | \( 1 + 6.06T + 59T^{2} \) |
| 61 | \( 1 + 5.75T + 61T^{2} \) |
| 67 | \( 1 + 2.28T + 67T^{2} \) |
| 71 | \( 1 + 9.05T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 - 2.29T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 + 3.51T + 97T^{2} \) |
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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
−7.74272871343575629151440546792, −7.39094633499065380514609699938, −6.43609002693575414798215896381, −5.46645791976008460479032624999, −4.56750321682397723713021791549, −4.13664886064607577318680563754, −3.07894010606610731420512072764, −2.51482737498361358783372886952, −1.47662289505368904168996900726, 0,
1.47662289505368904168996900726, 2.51482737498361358783372886952, 3.07894010606610731420512072764, 4.13664886064607577318680563754, 4.56750321682397723713021791549, 5.46645791976008460479032624999, 6.43609002693575414798215896381, 7.39094633499065380514609699938, 7.74272871343575629151440546792
