| L(s) = 1 | + 3·3-s − 4·7-s + 6·9-s + 3·11-s + 2·13-s − 3·17-s + 7·19-s − 12·21-s − 6·23-s + 9·27-s + 6·29-s + 10·31-s + 9·33-s + 6·39-s − 11·41-s + 4·43-s + 2·47-s + 9·49-s − 9·51-s + 14·53-s + 21·57-s + 4·59-s + 8·61-s − 24·63-s + 5·67-s − 18·69-s − 2·71-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.51·7-s + 2·9-s + 0.904·11-s + 0.554·13-s − 0.727·17-s + 1.60·19-s − 2.61·21-s − 1.25·23-s + 1.73·27-s + 1.11·29-s + 1.79·31-s + 1.56·33-s + 0.960·39-s − 1.71·41-s + 0.609·43-s + 0.291·47-s + 9/7·49-s − 1.26·51-s + 1.92·53-s + 2.78·57-s + 0.520·59-s + 1.02·61-s − 3.02·63-s + 0.610·67-s − 2.16·69-s − 0.237·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.393644300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.393644300\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−8.634387908231893384890208550953, −8.204765569990556202205094190182, −7.09495490440264169286928210927, −6.70951199133723358120764153947, −5.81926748120681599204450818323, −4.38918493583228095025199865943, −3.70761733220681653004768507952, −3.10375504611870128419524043499, −2.35420700729185270458758440312, −1.07536266970351529868491639943,
1.07536266970351529868491639943, 2.35420700729185270458758440312, 3.10375504611870128419524043499, 3.70761733220681653004768507952, 4.38918493583228095025199865943, 5.81926748120681599204450818323, 6.70951199133723358120764153947, 7.09495490440264169286928210927, 8.204765569990556202205094190182, 8.634387908231893384890208550953
