| L(s) = 1 | + 2-s − 2.30·3-s + 4-s − 2.30·6-s − 0.697·7-s + 8-s + 2.30·9-s + 2.60·11-s − 2.30·12-s − 6.30·13-s − 0.697·14-s + 16-s + 3.90·17-s + 2.30·18-s − 0.605·19-s + 1.60·21-s + 2.60·22-s − 1.69·23-s − 2.30·24-s − 6.30·26-s + 1.60·27-s − 0.697·28-s + 29-s − 7.90·31-s + 32-s − 6·33-s + 3.90·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.32·3-s + 0.5·4-s − 0.940·6-s − 0.263·7-s + 0.353·8-s + 0.767·9-s + 0.785·11-s − 0.664·12-s − 1.74·13-s − 0.186·14-s + 0.250·16-s + 0.947·17-s + 0.542·18-s − 0.138·19-s + 0.350·21-s + 0.555·22-s − 0.353·23-s − 0.470·24-s − 1.23·26-s + 0.308·27-s − 0.131·28-s + 0.185·29-s − 1.42·31-s + 0.176·32-s − 1.04·33-s + 0.670·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 2.30T + 3T^{2} \) |
| 7 | \( 1 + 0.697T + 7T^{2} \) |
| 11 | \( 1 - 2.60T + 11T^{2} \) |
| 13 | \( 1 + 6.30T + 13T^{2} \) |
| 17 | \( 1 - 3.90T + 17T^{2} \) |
| 19 | \( 1 + 0.605T + 19T^{2} \) |
| 23 | \( 1 + 1.69T + 23T^{2} \) |
| 31 | \( 1 + 7.90T + 31T^{2} \) |
| 37 | \( 1 + 9.81T + 37T^{2} \) |
| 41 | \( 1 + 8.60T + 41T^{2} \) |
| 43 | \( 1 + 3.30T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 3.90T + 53T^{2} \) |
| 59 | \( 1 + 0.908T + 59T^{2} \) |
| 61 | \( 1 + 3.09T + 61T^{2} \) |
| 67 | \( 1 - 9.21T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2.51T + 73T^{2} \) |
| 79 | \( 1 + 4.90T + 79T^{2} \) |
| 83 | \( 1 - 8.60T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 1.09T + 97T^{2} \) |
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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
−9.407990081614753518438997617253, −8.122977075012076541251019641000, −6.99003145307253373700181367280, −6.70473469805233267182206547619, −5.49718281366288437666415243477, −5.22680105345781307133436841878, −4.18060447012509745400533690702, −3.12032659178808281312473073020, −1.69465286739611247746851171454, 0,
1.69465286739611247746851171454, 3.12032659178808281312473073020, 4.18060447012509745400533690702, 5.22680105345781307133436841878, 5.49718281366288437666415243477, 6.70473469805233267182206547619, 6.99003145307253373700181367280, 8.122977075012076541251019641000, 9.407990081614753518438997617253
