| L(s) = 1 | − 3·5-s + 3·11-s − 4·13-s + 4·19-s + 4·25-s − 9·29-s + 31-s + 8·37-s + 10·43-s − 6·47-s + 3·53-s − 9·55-s + 3·59-s − 10·61-s + 12·65-s + 10·67-s − 6·71-s + 2·73-s + 79-s − 9·83-s − 6·89-s − 12·95-s − 97-s + 18·101-s − 8·103-s − 3·107-s + 14·109-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.904·11-s − 1.10·13-s + 0.917·19-s + 4/5·25-s − 1.67·29-s + 0.179·31-s + 1.31·37-s + 1.52·43-s − 0.875·47-s + 0.412·53-s − 1.21·55-s + 0.390·59-s − 1.28·61-s + 1.48·65-s + 1.22·67-s − 0.712·71-s + 0.234·73-s + 0.112·79-s − 0.987·83-s − 0.635·89-s − 1.23·95-s − 0.101·97-s + 1.79·101-s − 0.788·103-s − 0.290·107-s + 1.34·109-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−7.50870021842608327226913422365, −7.19740088877514161897699217101, −6.24531329037550765505333837146, −5.41909534255387024277594634459, −4.56688109344086501840116538407, −3.98572078920553132672263314204, −3.31044739848332951314922245100, −2.36338030198679401945614263937, −1.12166537142946599856459557708, 0,
1.12166537142946599856459557708, 2.36338030198679401945614263937, 3.31044739848332951314922245100, 3.98572078920553132672263314204, 4.56688109344086501840116538407, 5.41909534255387024277594634459, 6.24531329037550765505333837146, 7.19740088877514161897699217101, 7.50870021842608327226913422365
