| L(s) = 1 | − 3-s − 7-s + 9-s + 11-s + 2·13-s − 2·17-s + 21-s + 6·23-s − 27-s + 2·29-s − 6·31-s − 33-s + 2·37-s − 2·39-s − 8·43-s + 4·47-s + 49-s + 2·51-s + 6·53-s − 6·59-s − 8·61-s − 63-s + 10·67-s − 6·69-s − 6·73-s − 77-s + 16·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.485·17-s + 0.218·21-s + 1.25·23-s − 0.192·27-s + 0.371·29-s − 1.07·31-s − 0.174·33-s + 0.328·37-s − 0.320·39-s − 1.21·43-s + 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 0.781·59-s − 1.02·61-s − 0.125·63-s + 1.22·67-s − 0.722·69-s − 0.702·73-s − 0.113·77-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.784564026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.784564026\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.58425380350694, −12.00357151407068, −11.51691720203866, −11.20479301312921, −10.71458265723392, −10.37734590522338, −9.772835365410875, −9.364395555329247, −8.740598988588879, −8.675071955916477, −7.828383339152418, −7.341467257260915, −6.917943180441991, −6.477297154906892, −6.027158657430160, −5.571888230561487, −4.946024343126632, −4.619379081930823, −3.933270142615095, −3.478394480581786, −2.976190736731861, −2.256873782212980, −1.643630231519791, −1.013775438608719, −0.4081525300049825,
0.4081525300049825, 1.013775438608719, 1.643630231519791, 2.256873782212980, 2.976190736731861, 3.478394480581786, 3.933270142615095, 4.619379081930823, 4.946024343126632, 5.571888230561487, 6.027158657430160, 6.477297154906892, 6.917943180441991, 7.341467257260915, 7.828383339152418, 8.675071955916477, 8.740598988588879, 9.364395555329247, 9.772835365410875, 10.37734590522338, 10.71458265723392, 11.20479301312921, 11.51691720203866, 12.00357151407068, 12.58425380350694
