| L(s) = 1 | − 5-s − 7-s + 2·11-s − 5·13-s + 4·17-s − 5·19-s + 2·23-s + 25-s − 10·29-s − 8·31-s + 35-s − 3·37-s − 6·41-s + 4·43-s + 8·47-s − 6·49-s − 6·53-s − 2·55-s + 4·59-s − 5·61-s + 5·65-s − 7·67-s − 6·71-s − 9·73-s − 2·77-s + 3·79-s − 2·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.603·11-s − 1.38·13-s + 0.970·17-s − 1.14·19-s + 0.417·23-s + 1/5·25-s − 1.85·29-s − 1.43·31-s + 0.169·35-s − 0.493·37-s − 0.937·41-s + 0.609·43-s + 1.16·47-s − 6/7·49-s − 0.824·53-s − 0.269·55-s + 0.520·59-s − 0.640·61-s + 0.620·65-s − 0.855·67-s − 0.712·71-s − 1.05·73-s − 0.227·77-s + 0.337·79-s − 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−9.415262042209836561734202358047, −8.781918424918406976958800385400, −7.58167540835882906321030415541, −7.17367291154344014343165510267, −6.06076707656681085192149848012, −5.12766359400941907950130156696, −4.08509134025816421323180526965, −3.19323272439097456640445840897, −1.86399789194270921340618999768, 0,
1.86399789194270921340618999768, 3.19323272439097456640445840897, 4.08509134025816421323180526965, 5.12766359400941907950130156696, 6.06076707656681085192149848012, 7.17367291154344014343165510267, 7.58167540835882906321030415541, 8.781918424918406976958800385400, 9.415262042209836561734202358047
