| L(s) = 1 | + 2·3-s − 4·5-s + 7-s + 9-s − 8·15-s − 2·17-s − 2·19-s + 2·21-s + 8·23-s + 11·25-s − 4·27-s + 2·29-s + 4·31-s − 4·35-s − 6·37-s − 2·41-s + 8·43-s − 4·45-s − 4·47-s + 49-s − 4·51-s − 10·53-s − 4·57-s + 6·59-s + 4·61-s + 63-s − 12·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s + 0.377·7-s + 1/3·9-s − 2.06·15-s − 0.485·17-s − 0.458·19-s + 0.436·21-s + 1.66·23-s + 11/5·25-s − 0.769·27-s + 0.371·29-s + 0.718·31-s − 0.676·35-s − 0.986·37-s − 0.312·41-s + 1.21·43-s − 0.596·45-s − 0.583·47-s + 1/7·49-s − 0.560·51-s − 1.37·53-s − 0.529·57-s + 0.781·59-s + 0.512·61-s + 0.125·63-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9480191156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9480191156\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.16780114648984000281229819869, −14.47141648264651804115718476828, −13.10747313979957160372604181414, −11.85083361056233693149695469963, −10.85093898417625194511634818884, −8.928329697232844766585907221767, −8.188057214509100026281306554311, −7.14438434722230698367292292955, −4.50185552371175935345657331179, −3.14137792992390816749783057891,
3.14137792992390816749783057891, 4.50185552371175935345657331179, 7.14438434722230698367292292955, 8.188057214509100026281306554311, 8.928329697232844766585907221767, 10.85093898417625194511634818884, 11.85083361056233693149695469963, 13.10747313979957160372604181414, 14.47141648264651804115718476828, 15.16780114648984000281229819869
