| L(s) = 1 | + 2·5-s + 7-s − 3·9-s + 2·13-s − 17-s + 4·19-s − 25-s + 6·29-s + 2·35-s − 6·37-s + 6·41-s − 12·43-s − 6·45-s − 8·47-s + 49-s − 2·53-s − 4·59-s − 2·61-s − 3·63-s + 4·65-s − 12·67-s − 2·73-s − 8·79-s + 9·81-s + 12·83-s − 2·85-s + 10·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s − 9-s + 0.554·13-s − 0.242·17-s + 0.917·19-s − 1/5·25-s + 1.11·29-s + 0.338·35-s − 0.986·37-s + 0.937·41-s − 1.82·43-s − 0.894·45-s − 1.16·47-s + 1/7·49-s − 0.274·53-s − 0.520·59-s − 0.256·61-s − 0.377·63-s + 0.496·65-s − 1.46·67-s − 0.234·73-s − 0.900·79-s + 81-s + 1.31·83-s − 0.216·85-s + 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.401632176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.401632176\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−13.18537969865487, −12.29778434584820, −12.00804905613585, −11.54340086204775, −11.08825545475554, −10.60434263755947, −10.13539622576729, −9.668738359963593, −9.151640853106909, −8.747287624922898, −8.241010185841017, −7.871780137820728, −7.207886424156419, −6.573942542974825, −6.209106126168191, −5.733944093624923, −5.223997617550042, −4.832555640237057, −4.196045009273095, −3.305411096034023, −3.126397789972566, −2.393073108549677, −1.739444548881112, −1.316258856839234, −0.4185141785103134,
0.4185141785103134, 1.316258856839234, 1.739444548881112, 2.393073108549677, 3.126397789972566, 3.305411096034023, 4.196045009273095, 4.832555640237057, 5.223997617550042, 5.733944093624923, 6.209106126168191, 6.573942542974825, 7.207886424156419, 7.871780137820728, 8.241010185841017, 8.747287624922898, 9.151640853106909, 9.668738359963593, 10.13539622576729, 10.60434263755947, 11.08825545475554, 11.54340086204775, 12.00804905613585, 12.29778434584820, 13.18537969865487
