L(s) = 1 | + 3-s + 2·5-s + 2·7-s + 9-s + 4·11-s − 6·13-s + 2·15-s + 6·17-s + 6·19-s + 2·21-s − 25-s + 27-s − 6·29-s + 8·31-s + 4·33-s + 4·35-s + 8·37-s − 6·39-s − 6·41-s − 10·43-s + 2·45-s + 4·47-s − 3·49-s + 6·51-s − 2·53-s + 8·55-s + 6·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s + 0.516·15-s + 1.45·17-s + 1.37·19-s + 0.436·21-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.696·33-s + 0.676·35-s + 1.31·37-s − 0.960·39-s − 0.937·41-s − 1.52·43-s + 0.298·45-s + 0.583·47-s − 3/7·49-s + 0.840·51-s − 0.274·53-s + 1.07·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.285824650\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.285824650\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−16.59533518287548, −15.61762927673653, −14.81019541516849, −14.55489111660723, −14.25667173586826, −13.53470247139311, −13.10258522857955, −12.04062960129568, −11.92576591936095, −11.33789062719572, −10.10463926418835, −9.918708214200294, −9.504859783993659, −8.825180599265227, −7.894247331557890, −7.657274134965222, −6.865414976922011, −6.165344676683126, −5.246400042389670, −5.009799029245355, −3.989135278291676, −3.252879146249315, −2.434832528188577, −1.693327154735086, −0.9790974914612601,
0.9790974914612601, 1.693327154735086, 2.434832528188577, 3.252879146249315, 3.989135278291676, 5.009799029245355, 5.246400042389670, 6.165344676683126, 6.865414976922011, 7.657274134965222, 7.894247331557890, 8.825180599265227, 9.504859783993659, 9.918708214200294, 10.10463926418835, 11.33789062719572, 11.92576591936095, 12.04062960129568, 13.10258522857955, 13.53470247139311, 14.25667173586826, 14.55489111660723, 14.81019541516849, 15.61762927673653, 16.59533518287548