| L(s) = 1 | + 3-s − 2·5-s + 2·7-s − 2·9-s + 2·11-s + 7·13-s − 2·15-s − 3·17-s − 5·19-s + 2·21-s − 3·23-s − 25-s − 5·27-s − 9·29-s − 8·31-s + 2·33-s − 4·35-s + 3·37-s + 7·39-s − 2·41-s + 4·43-s + 4·45-s − 10·47-s − 3·49-s − 3·51-s − 4·55-s − 5·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.755·7-s − 2/3·9-s + 0.603·11-s + 1.94·13-s − 0.516·15-s − 0.727·17-s − 1.14·19-s + 0.436·21-s − 0.625·23-s − 1/5·25-s − 0.962·27-s − 1.67·29-s − 1.43·31-s + 0.348·33-s − 0.676·35-s + 0.493·37-s + 1.12·39-s − 0.312·41-s + 0.609·43-s + 0.596·45-s − 1.45·47-s − 3/7·49-s − 0.420·51-s − 0.539·55-s − 0.662·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 179776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6101003662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6101003662\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 53 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−13.33385617939707, −12.73939132102157, −12.17540040587572, −11.56312425067440, −11.23056279102492, −10.96447872487766, −10.66109899750049, −9.608724791904024, −9.199860625752720, −8.800374889524046, −8.332652408642186, −8.014557118188957, −7.606011892140948, −6.920700716406760, −6.224235705430795, −5.995698326081371, −5.339838494575747, −4.571061207017261, −3.969322928330847, −3.787542710850776, −3.290851482330607, −2.428794223286687, −1.721003206042151, −1.466452263732423, −0.2042990216628368,
0.2042990216628368, 1.466452263732423, 1.721003206042151, 2.428794223286687, 3.290851482330607, 3.787542710850776, 3.969322928330847, 4.571061207017261, 5.339838494575747, 5.995698326081371, 6.224235705430795, 6.920700716406760, 7.606011892140948, 8.014557118188957, 8.332652408642186, 8.800374889524046, 9.199860625752720, 9.608724791904024, 10.66109899750049, 10.96447872487766, 11.23056279102492, 11.56312425067440, 12.17540040587572, 12.73939132102157, 13.33385617939707
