| L(s) = 1 | − 3-s + 2·7-s − 2·9-s − 13-s + 4·17-s − 4·19-s − 2·21-s − 23-s + 5·27-s − 3·29-s − 31-s + 8·37-s + 39-s − 5·41-s + 6·43-s − 9·47-s − 3·49-s − 4·51-s − 2·53-s + 4·57-s − 4·63-s − 4·67-s + 69-s + 3·71-s − 7·73-s + 4·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s − 2/3·9-s − 0.277·13-s + 0.970·17-s − 0.917·19-s − 0.436·21-s − 0.208·23-s + 0.962·27-s − 0.557·29-s − 0.179·31-s + 1.31·37-s + 0.160·39-s − 0.780·41-s + 0.914·43-s − 1.31·47-s − 3/7·49-s − 0.560·51-s − 0.274·53-s + 0.529·57-s − 0.503·63-s − 0.488·67-s + 0.120·69-s + 0.356·71-s − 0.819·73-s + 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−8.026104839206072484718820015734, −7.28357641965222627403634626356, −6.33573760181621128718429717667, −5.76751732391269470885301891120, −5.04777616072826196553892724666, −4.37223386482384364604870545239, −3.33910206209539937228556444848, −2.37686439291735034085386510972, −1.31679312432706868249748039940, 0,
1.31679312432706868249748039940, 2.37686439291735034085386510972, 3.33910206209539937228556444848, 4.37223386482384364604870545239, 5.04777616072826196553892724666, 5.76751732391269470885301891120, 6.33573760181621128718429717667, 7.28357641965222627403634626356, 8.026104839206072484718820015734
