| L(s) = 1 | − 3·3-s − 7-s + 6·9-s − 6·11-s + 13-s + 3·21-s + 23-s − 5·25-s − 9·27-s − 3·29-s − 3·31-s + 18·33-s − 8·37-s − 3·39-s + 9·41-s + 4·43-s + 13·47-s + 49-s + 4·53-s + 4·59-s + 2·61-s − 6·63-s − 4·67-s − 3·69-s − 5·71-s + 3·73-s + 15·75-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.377·7-s + 2·9-s − 1.80·11-s + 0.277·13-s + 0.654·21-s + 0.208·23-s − 25-s − 1.73·27-s − 0.557·29-s − 0.538·31-s + 3.13·33-s − 1.31·37-s − 0.480·39-s + 1.40·41-s + 0.609·43-s + 1.89·47-s + 1/7·49-s + 0.549·53-s + 0.520·59-s + 0.256·61-s − 0.755·63-s − 0.488·67-s − 0.361·69-s − 0.593·71-s + 0.351·73-s + 1.73·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5435399033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5435399033\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−10.00721660086992619398563740451, −8.987759056977041645209445829409, −7.70534404497972910768407911666, −7.18348608212619323873685524819, −6.04063030407283117353279123819, −5.59566766322433232164217084851, −4.85135202660753106632850902635, −3.74886735085206739000907003928, −2.25309964306896007792618447153, −0.56804643533526679075759024976,
0.56804643533526679075759024976, 2.25309964306896007792618447153, 3.74886735085206739000907003928, 4.85135202660753106632850902635, 5.59566766322433232164217084851, 6.04063030407283117353279123819, 7.18348608212619323873685524819, 7.70534404497972910768407911666, 8.987759056977041645209445829409, 10.00721660086992619398563740451
