| L(s) = 1 | − 4·7-s + 4·11-s − 13-s + 2·17-s + 10·29-s − 4·31-s + 2·37-s − 6·41-s − 12·43-s + 9·49-s + 6·53-s + 12·59-s − 2·61-s − 8·67-s − 2·73-s − 16·77-s − 8·79-s − 4·83-s + 2·89-s + 4·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 8·119-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1.20·11-s − 0.277·13-s + 0.485·17-s + 1.85·29-s − 0.718·31-s + 0.328·37-s − 0.937·41-s − 1.82·43-s + 9/7·49-s + 0.824·53-s + 1.56·59-s − 0.256·61-s − 0.977·67-s − 0.234·73-s − 1.82·77-s − 0.900·79-s − 0.439·83-s + 0.211·89-s + 0.419·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.733·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.75859499202109, −14.42411154772300, −13.75899233536595, −13.29697691936743, −12.84075962532412, −12.21385809623017, −11.84390293864844, −11.46124814464235, −10.41031453840180, −10.18756428404471, −9.694218947993799, −9.128163101218480, −8.651968496854667, −8.078281701434993, −7.198110086251223, −6.745188623544328, −6.462500634744769, −5.774193023517916, −5.145902072596059, −4.354564554568270, −3.762592238951736, −3.202530659678546, −2.691456692193998, −1.703542873373594, −0.9195064949044537, 0,
0.9195064949044537, 1.703542873373594, 2.691456692193998, 3.202530659678546, 3.762592238951736, 4.354564554568270, 5.145902072596059, 5.774193023517916, 6.462500634744769, 6.745188623544328, 7.198110086251223, 8.078281701434993, 8.651968496854667, 9.128163101218480, 9.694218947993799, 10.18756428404471, 10.41031453840180, 11.46124814464235, 11.84390293864844, 12.21385809623017, 12.84075962532412, 13.29697691936743, 13.75899233536595, 14.42411154772300, 14.75859499202109
