| L(s) = 1 | − 3-s + 5-s − 4·7-s + 9-s − 4·11-s − 2·13-s − 15-s + 2·17-s − 2·19-s + 4·21-s − 6·23-s + 25-s − 27-s + 6·29-s − 10·31-s + 4·33-s − 4·35-s + 37-s + 2·39-s − 2·41-s − 2·43-s + 45-s + 4·47-s + 9·49-s − 2·51-s − 10·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.458·19-s + 0.872·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.79·31-s + 0.696·33-s − 0.676·35-s + 0.164·37-s + 0.320·39-s − 0.312·41-s − 0.304·43-s + 0.149·45-s + 0.583·47-s + 9/7·49-s − 0.280·51-s − 1.37·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−15.75597482663717, −14.92344569816459, −14.44696952191094, −13.62874317000134, −13.40813685820865, −12.62733110784961, −12.48259836637186, −12.00583259246071, −11.06369265822480, −10.60005947036310, −10.14591170557370, −9.698675127699775, −9.292420958170938, −8.447346357451370, −7.833631394748123, −7.197054741012743, −6.706196231110890, −5.926008602926275, −5.800270203374429, −5.005888224846858, −4.363299914600806, −3.534756907116056, −2.913550307911601, −2.309024843626567, −1.410201346196896, 0, 0,
1.410201346196896, 2.309024843626567, 2.913550307911601, 3.534756907116056, 4.363299914600806, 5.005888224846858, 5.800270203374429, 5.926008602926275, 6.706196231110890, 7.197054741012743, 7.833631394748123, 8.447346357451370, 9.292420958170938, 9.698675127699775, 10.14591170557370, 10.60005947036310, 11.06369265822480, 12.00583259246071, 12.48259836637186, 12.62733110784961, 13.40813685820865, 13.62874317000134, 14.44696952191094, 14.92344569816459, 15.75597482663717
