L(s) = 1 | − 3-s + 2·5-s + 7-s + 9-s − 4·11-s − 2·13-s − 2·15-s + 17-s − 8·19-s − 21-s − 25-s − 27-s + 2·29-s − 8·31-s + 4·33-s + 2·35-s + 2·37-s + 2·39-s − 6·41-s + 8·43-s + 2·45-s − 8·47-s + 49-s − 51-s + 6·53-s − 8·55-s + 8·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.516·15-s + 0.242·17-s − 1.83·19-s − 0.218·21-s − 1/5·25-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.696·33-s + 0.338·35-s + 0.328·37-s + 0.320·39-s − 0.937·41-s + 1.21·43-s + 0.298·45-s − 1.16·47-s + 1/7·49-s − 0.140·51-s + 0.824·53-s − 1.07·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11424 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.330564220\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.330564220\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−16.62159792228843, −15.97109115587702, −15.21333635542709, −14.81836617197687, −14.19915891095805, −13.45048906543703, −12.99801513754506, −12.55822802206837, −11.89037442813184, −11.06982185763235, −10.65164902744626, −10.15826910587859, −9.589302739708166, −8.815799284730997, −8.153844676967991, −7.507857101270178, −6.810980783790409, −6.062948280479294, −5.579974105704396, −4.939632289743711, −4.350225957410654, −3.317824175263968, −2.189000811640283, −1.957740457476233, −0.5205477587777299,
0.5205477587777299, 1.957740457476233, 2.189000811640283, 3.317824175263968, 4.350225957410654, 4.939632289743711, 5.579974105704396, 6.062948280479294, 6.810980783790409, 7.507857101270178, 8.153844676967991, 8.815799284730997, 9.589302739708166, 10.15826910587859, 10.65164902744626, 11.06982185763235, 11.89037442813184, 12.55822802206837, 12.99801513754506, 13.45048906543703, 14.19915891095805, 14.81836617197687, 15.21333635542709, 15.97109115587702, 16.62159792228843