| L(s) = 1 | + 3-s + 9-s − 4·11-s − 2·13-s − 2·17-s + 4·19-s + 27-s + 2·29-s − 4·33-s − 10·37-s − 2·39-s + 10·41-s − 4·43-s + 8·47-s − 7·49-s − 2·51-s − 10·53-s + 4·57-s − 4·59-s + 2·61-s − 12·67-s + 8·71-s − 10·73-s + 81-s − 12·83-s + 2·87-s − 6·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 0.192·27-s + 0.371·29-s − 0.696·33-s − 1.64·37-s − 0.320·39-s + 1.56·41-s − 0.609·43-s + 1.16·47-s − 49-s − 0.280·51-s − 1.37·53-s + 0.529·57-s − 0.520·59-s + 0.256·61-s − 1.46·67-s + 0.949·71-s − 1.17·73-s + 1/9·81-s − 1.31·83-s + 0.214·87-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 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
−7.84465462281701093084420479537, −7.39546104131459972836674622338, −6.60942929553535216154216963636, −5.59929057414059862389948014289, −4.97287019204270070424993179755, −4.17881549969972779526027182436, −3.10526976657196724130917180316, −2.58465419848810939203642390782, −1.51207711394055808445468187309, 0,
1.51207711394055808445468187309, 2.58465419848810939203642390782, 3.10526976657196724130917180316, 4.17881549969972779526027182436, 4.97287019204270070424993179755, 5.59929057414059862389948014289, 6.60942929553535216154216963636, 7.39546104131459972836674622338, 7.84465462281701093084420479537
