| L(s) = 1 | + 2·5-s − 2·7-s − 3·9-s − 11-s − 2·13-s + 6·17-s − 19-s + 8·23-s − 25-s − 6·29-s − 6·31-s − 4·35-s + 8·37-s + 6·41-s + 8·43-s − 6·45-s + 8·47-s − 3·49-s + 12·53-s − 2·55-s − 8·61-s + 6·63-s − 4·65-s + 8·67-s + 6·71-s − 14·73-s + 2·77-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 9-s − 0.301·11-s − 0.554·13-s + 1.45·17-s − 0.229·19-s + 1.66·23-s − 1/5·25-s − 1.11·29-s − 1.07·31-s − 0.676·35-s + 1.31·37-s + 0.937·41-s + 1.21·43-s − 0.894·45-s + 1.16·47-s − 3/7·49-s + 1.64·53-s − 0.269·55-s − 1.02·61-s + 0.755·63-s − 0.496·65-s + 0.977·67-s + 0.712·71-s − 1.63·73-s + 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.740022963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.740022963\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−8.873549565618843688615516774125, −7.71683031669699780209908431978, −7.25642256397993753530881147655, −6.09679106383446582994345161763, −5.72435322567187799246362343196, −5.06574132185716481191903236655, −3.77071146397316569049713132542, −2.91767706694427689436347518916, −2.23165447040934725079950854368, −0.76944286488840743808641098222,
0.76944286488840743808641098222, 2.23165447040934725079950854368, 2.91767706694427689436347518916, 3.77071146397316569049713132542, 5.06574132185716481191903236655, 5.72435322567187799246362343196, 6.09679106383446582994345161763, 7.25642256397993753530881147655, 7.71683031669699780209908431978, 8.873549565618843688615516774125
