| L(s) = 1 | − 2.57·2-s − 0.611·3-s + 4.62·4-s + 5-s + 1.57·6-s + 7-s − 6.75·8-s − 2.62·9-s − 2.57·10-s − 2.82·12-s − 4.82·13-s − 2.57·14-s − 0.611·15-s + 8.14·16-s + 1.49·17-s + 6.75·18-s − 2.66·19-s + 4.62·20-s − 0.611·21-s + 2.18·23-s + 4.13·24-s + 25-s + 12.4·26-s + 3.44·27-s + 4.62·28-s − 7.92·29-s + 1.57·30-s + ⋯ |
| L(s) = 1 | − 1.82·2-s − 0.353·3-s + 2.31·4-s + 0.447·5-s + 0.642·6-s + 0.377·7-s − 2.38·8-s − 0.875·9-s − 0.814·10-s − 0.816·12-s − 1.33·13-s − 0.687·14-s − 0.157·15-s + 2.03·16-s + 0.361·17-s + 1.59·18-s − 0.611·19-s + 1.03·20-s − 0.133·21-s + 0.455·23-s + 0.843·24-s + 0.200·25-s + 2.43·26-s + 0.662·27-s + 0.874·28-s − 1.47·29-s + 0.287·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.57T + 2T^{2} \) |
| 3 | \( 1 + 0.611T + 3T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 - 1.49T + 17T^{2} \) |
| 19 | \( 1 + 2.66T + 19T^{2} \) |
| 23 | \( 1 - 2.18T + 23T^{2} \) |
| 29 | \( 1 + 7.92T + 29T^{2} \) |
| 31 | \( 1 + 2.91T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 - 4.16T + 41T^{2} \) |
| 43 | \( 1 - 6.55T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 2.74T + 59T^{2} \) |
| 61 | \( 1 - 7.84T + 61T^{2} \) |
| 67 | \( 1 + 4.55T + 67T^{2} \) |
| 71 | \( 1 + 3.49T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 + 6.77T + 79T^{2} \) |
| 83 | \( 1 - 7.37T + 83T^{2} \) |
| 89 | \( 1 + 8.51T + 89T^{2} \) |
| 97 | \( 1 + 7.78T + 97T^{2} \) |
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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
−8.048780628287479790039067529041, −7.49422441228454899460188460138, −6.91712253669969218400250252430, −5.88134342197840583288112341902, −5.50406114438484283290313240191, −4.24945398423167040421829769516, −2.69987753129049006418441844344, −2.30223557204129366102825329663, −1.08981119105473520354002721672, 0,
1.08981119105473520354002721672, 2.30223557204129366102825329663, 2.69987753129049006418441844344, 4.24945398423167040421829769516, 5.50406114438484283290313240191, 5.88134342197840583288112341902, 6.91712253669969218400250252430, 7.49422441228454899460188460138, 8.048780628287479790039067529041
