L(s) = 1 | + 3-s − 2·5-s − 2·7-s + 9-s + 2·11-s − 2·13-s − 2·15-s − 4·17-s − 2·21-s + 23-s − 25-s + 27-s − 10·29-s + 2·33-s + 4·35-s − 4·37-s − 2·39-s − 6·41-s + 4·43-s − 2·45-s − 8·47-s − 3·49-s − 4·51-s − 6·53-s − 4·55-s − 4·59-s + 8·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 0.516·15-s − 0.970·17-s − 0.436·21-s + 0.208·23-s − 1/5·25-s + 0.192·27-s − 1.85·29-s + 0.348·33-s + 0.676·35-s − 0.657·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.298·45-s − 1.16·47-s − 3/7·49-s − 0.560·51-s − 0.824·53-s − 0.539·55-s − 0.520·59-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.351400696245569298459028256207, −8.697349756574051915031753577483, −7.74045844024391827741185861892, −7.06743204112300994256415402420, −6.25199234905726004811235315378, −4.92685183265423845819251762023, −3.91973999755727036502453272135, −3.27308383966267084098272599041, −1.95980482343429025156607292353, 0,
1.95980482343429025156607292353, 3.27308383966267084098272599041, 3.91973999755727036502453272135, 4.92685183265423845819251762023, 6.25199234905726004811235315378, 7.06743204112300994256415402420, 7.74045844024391827741185861892, 8.697349756574051915031753577483, 9.351400696245569298459028256207