| L(s) = 1 | − 2·5-s + 2·11-s − 4·13-s − 6·17-s − 8·19-s − 6·23-s − 25-s + 10·29-s − 4·31-s + 6·37-s + 6·41-s − 4·43-s + 8·47-s − 2·53-s − 4·55-s − 4·59-s − 8·61-s + 8·65-s + 8·67-s − 10·71-s + 4·73-s − 4·79-s + 12·83-s + 12·85-s + 14·89-s + 16·95-s + 4·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.603·11-s − 1.10·13-s − 1.45·17-s − 1.83·19-s − 1.25·23-s − 1/5·25-s + 1.85·29-s − 0.718·31-s + 0.986·37-s + 0.937·41-s − 0.609·43-s + 1.16·47-s − 0.274·53-s − 0.539·55-s − 0.520·59-s − 1.02·61-s + 0.992·65-s + 0.977·67-s − 1.18·71-s + 0.468·73-s − 0.450·79-s + 1.31·83-s + 1.30·85-s + 1.48·89-s + 1.64·95-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8116985192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8116985192\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−7.899113846890409665577011513332, −7.33617816455226293213945347049, −6.43760935956942238381405409471, −6.14421070646052528445807958869, −4.77041920554517839461286789177, −4.37455115973192812140463707191, −3.79766730312154259979498127992, −2.59150709519871201789294173801, −1.98176544333354037351104687406, −0.43333167368032222610975469437,
0.43333167368032222610975469437, 1.98176544333354037351104687406, 2.59150709519871201789294173801, 3.79766730312154259979498127992, 4.37455115973192812140463707191, 4.77041920554517839461286789177, 6.14421070646052528445807958869, 6.43760935956942238381405409471, 7.33617816455226293213945347049, 7.899113846890409665577011513332
