| L(s) = 1 | + 5·2-s + 17·4-s + 45·8-s + 68·11-s + 89·16-s + 340·22-s + 40·23-s − 125·25-s + 166·29-s + 85·32-s + 450·37-s − 180·43-s + 1.15e3·44-s + 200·46-s − 625·50-s − 590·53-s + 830·58-s − 287·64-s − 740·67-s − 688·71-s + 2.25e3·74-s − 1.38e3·79-s − 900·86-s + 3.06e3·88-s + 680·92-s − 2.12e3·100-s − 2.95e3·106-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 17/8·4-s + 1.98·8-s + 1.86·11-s + 1.39·16-s + 3.29·22-s + 0.362·23-s − 25-s + 1.06·29-s + 0.469·32-s + 1.99·37-s − 0.638·43-s + 3.96·44-s + 0.641·46-s − 1.76·50-s − 1.52·53-s + 1.87·58-s − 0.560·64-s − 1.34·67-s − 1.15·71-s + 3.53·74-s − 1.97·79-s − 1.12·86-s + 3.70·88-s + 0.770·92-s − 2.12·100-s − 2.70·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.197291470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.197291470\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 5 T + p^{3} T^{2} \) |
| 5 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 - 68 T + p^{3} T^{2} \) |
| 13 | \( 1 + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 + p^{3} T^{2} \) |
| 23 | \( 1 - 40 T + p^{3} T^{2} \) |
| 29 | \( 1 - 166 T + p^{3} T^{2} \) |
| 31 | \( 1 + p^{3} T^{2} \) |
| 37 | \( 1 - 450 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 + 180 T + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + 590 T + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 + p^{3} T^{2} \) |
| 67 | \( 1 + 740 T + p^{3} T^{2} \) |
| 71 | \( 1 + 688 T + p^{3} T^{2} \) |
| 73 | \( 1 + p^{3} T^{2} \) |
| 79 | \( 1 + 1384 T + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 + p^{3} T^{2} \) |
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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
−11.31435525835004719778360066343, −9.983927217498222619537261635393, −8.919698787967461541489595300935, −7.54254320009823258691593577599, −6.49740437123140591846158446986, −5.99316120484815613506263511199, −4.67113003251447696092695034138, −3.98515496184856310144687655078, −2.92446605741448292310787102731, −1.48976179735318305341387801246,
1.48976179735318305341387801246, 2.92446605741448292310787102731, 3.98515496184856310144687655078, 4.67113003251447696092695034138, 5.99316120484815613506263511199, 6.49740437123140591846158446986, 7.54254320009823258691593577599, 8.919698787967461541489595300935, 9.983927217498222619537261635393, 11.31435525835004719778360066343
