| L(s) = 1 | + 1.73·3-s + 3.46·5-s − 3·7-s − 5·11-s + 5.99·15-s + 7·17-s − 5·19-s − 5.19·21-s − 5.19·23-s + 6.99·25-s − 5.19·27-s − 5·29-s − 2·31-s − 8.66·33-s − 10.3·35-s − 5.19·37-s − 1.73·41-s − 5.19·43-s + 4·47-s + 2·49-s + 12.1·51-s + 4·53-s − 17.3·55-s − 8.66·57-s − 7·59-s + 3·61-s + ⋯ |
| L(s) = 1 | + 1.00·3-s + 1.54·5-s − 1.13·7-s − 1.50·11-s + 1.54·15-s + 1.69·17-s − 1.14·19-s − 1.13·21-s − 1.08·23-s + 1.39·25-s − 1.00·27-s − 0.928·29-s − 0.359·31-s − 1.50·33-s − 1.75·35-s − 0.854·37-s − 0.270·41-s − 0.792·43-s + 0.583·47-s + 0.285·49-s + 1.69·51-s + 0.549·53-s − 2.33·55-s − 1.14·57-s − 0.911·59-s + 0.384·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 1.73T + 3T^{2} \) |
| 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 17 | \( 1 - 7T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + 5.19T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 5.19T + 37T^{2} \) |
| 41 | \( 1 + 1.73T + 41T^{2} \) |
| 43 | \( 1 + 5.19T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 7T + 59T^{2} \) |
| 61 | \( 1 - 3T + 61T^{2} \) |
| 67 | \( 1 - 3T + 67T^{2} \) |
| 71 | \( 1 + 7T + 71T^{2} \) |
| 73 | \( 1 + 3.46T + 73T^{2} \) |
| 79 | \( 1 - 3.46T + 79T^{2} \) |
| 83 | \( 1 - 14T + 83T^{2} \) |
| 89 | \( 1 + 1.73T + 89T^{2} \) |
| 97 | \( 1 + 8.66T + 97T^{2} \) |
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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
−7.937985447532870769502708935694, −7.18555194996690999610474164184, −6.20036243603643610623908836777, −5.72714835900501338984956196883, −5.16255748131257740172559120024, −3.77023152874057563501130278683, −3.09529316933892911448125077077, −2.41555511615693334495499462860, −1.76752275129678328409173856115, 0,
1.76752275129678328409173856115, 2.41555511615693334495499462860, 3.09529316933892911448125077077, 3.77023152874057563501130278683, 5.16255748131257740172559120024, 5.72714835900501338984956196883, 6.20036243603643610623908836777, 7.18555194996690999610474164184, 7.937985447532870769502708935694
