| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 4·11-s − 2·13-s − 15-s + 2·17-s − 4·19-s − 21-s + 8·23-s + 25-s − 27-s − 2·29-s − 4·33-s + 35-s + 6·37-s + 2·39-s − 6·41-s + 4·43-s + 45-s + 49-s − 2·51-s − 10·53-s + 4·55-s + 4·57-s − 12·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s − 0.218·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.696·33-s + 0.169·35-s + 0.986·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s + 1/7·49-s − 0.280·51-s − 1.37·53-s + 0.539·55-s + 0.529·57-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.723611497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.723611497\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.403231692036894201001461624609, −8.697411889244989128372525290898, −7.66810120881612925960935375865, −6.79343333191751589242430738256, −6.21460460929070095255225769494, −5.21358590963430668330436186572, −4.54298127009285179950723294323, −3.45502491861020799705897381013, −2.11961083786733012234885915373, −0.991396209380100426279606276465,
0.991396209380100426279606276465, 2.11961083786733012234885915373, 3.45502491861020799705897381013, 4.54298127009285179950723294323, 5.21358590963430668330436186572, 6.21460460929070095255225769494, 6.79343333191751589242430738256, 7.66810120881612925960935375865, 8.697411889244989128372525290898, 9.403231692036894201001461624609
