| L(s) = 1 | + 3-s + 5-s − 2·9-s + 6·11-s + 4·13-s + 15-s + 2·19-s + 3·23-s + 25-s − 5·27-s − 3·29-s + 8·31-s + 6·33-s − 4·37-s + 4·39-s − 9·41-s + 7·43-s − 2·45-s − 6·53-s + 6·55-s + 2·57-s − 6·59-s − 5·61-s + 4·65-s − 5·67-s + 3·69-s + 6·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 2/3·9-s + 1.80·11-s + 1.10·13-s + 0.258·15-s + 0.458·19-s + 0.625·23-s + 1/5·25-s − 0.962·27-s − 0.557·29-s + 1.43·31-s + 1.04·33-s − 0.657·37-s + 0.640·39-s − 1.40·41-s + 1.06·43-s − 0.298·45-s − 0.824·53-s + 0.809·55-s + 0.264·57-s − 0.781·59-s − 0.640·61-s + 0.496·65-s − 0.610·67-s + 0.361·69-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.981430920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.981430920\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.585611469505637779917975201697, −7.922123577593787592042594371067, −6.84529302039872832385406815476, −6.32452963481278771786211763193, −5.63095877070288112514665916724, −4.60530874348065801098420211527, −3.63703389202489724923396682704, −3.13138504872146264156164214775, −1.92318340352558472695927844813, −1.04621856322269760220444445166,
1.04621856322269760220444445166, 1.92318340352558472695927844813, 3.13138504872146264156164214775, 3.63703389202489724923396682704, 4.60530874348065801098420211527, 5.63095877070288112514665916724, 6.32452963481278771786211763193, 6.84529302039872832385406815476, 7.922123577593787592042594371067, 8.585611469505637779917975201697
