| L(s) = 1 | − 3-s + 2·5-s + 9-s − 4·11-s + 2·13-s − 2·15-s − 17-s + 4·19-s − 8·23-s − 25-s − 27-s + 6·29-s + 4·33-s − 2·37-s − 2·39-s − 10·41-s + 4·43-s + 2·45-s + 51-s + 6·53-s − 8·55-s − 4·57-s − 4·59-s − 6·61-s + 4·65-s + 12·67-s + 8·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.516·15-s − 0.242·17-s + 0.917·19-s − 1.66·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.696·33-s − 0.328·37-s − 0.320·39-s − 1.56·41-s + 0.609·43-s + 0.298·45-s + 0.140·51-s + 0.824·53-s − 1.07·55-s − 0.529·57-s − 0.520·59-s − 0.768·61-s + 0.496·65-s + 1.46·67-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−15.34278464074561, −14.25035676189072, −13.80061354795741, −13.73424537956293, −12.87493786554776, −12.59179493632602, −11.70893206830606, −11.61233451156987, −10.59702146784715, −10.40305958589930, −9.883775573448741, −9.417055783835954, −8.588542450820088, −8.109404455843919, −7.550933768295637, −6.828163352613426, −6.227063057215252, −5.789829411195115, −5.249195005231970, −4.764561200297854, −3.918346066480998, −3.235624141554556, −2.373787105739376, −1.871690264398268, −0.9676872054866997, 0,
0.9676872054866997, 1.871690264398268, 2.373787105739376, 3.235624141554556, 3.918346066480998, 4.764561200297854, 5.249195005231970, 5.789829411195115, 6.227063057215252, 6.828163352613426, 7.550933768295637, 8.109404455843919, 8.588542450820088, 9.417055783835954, 9.883775573448741, 10.40305958589930, 10.59702146784715, 11.61233451156987, 11.70893206830606, 12.59179493632602, 12.87493786554776, 13.73424537956293, 13.80061354795741, 14.25035676189072, 15.34278464074561
