L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 4·11-s − 2·13-s − 15-s − 2·17-s − 21-s + 25-s − 27-s + 6·29-s + 8·31-s − 4·33-s + 35-s + 2·37-s + 2·39-s − 6·41-s + 4·43-s + 45-s + 49-s + 2·51-s − 6·53-s + 4·55-s + 8·59-s − 2·61-s + 63-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s − 0.485·17-s − 0.218·21-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.696·33-s + 0.169·35-s + 0.328·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s + 1/7·49-s + 0.280·51-s − 0.824·53-s + 0.539·55-s + 1.04·59-s − 0.256·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.869666743\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.869666743\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.656168874052238393920378245841, −7.897537864202885842321759356001, −6.83346147766309879590035918294, −6.50084583917956462186113244261, −5.62767145405384596348877496208, −4.74006328207835795732614245125, −4.20688340878182674339931011174, −2.96991644296420276801905236432, −1.89255984307424194789110869240, −0.874889815084987039484755102120,
0.874889815084987039484755102120, 1.89255984307424194789110869240, 2.96991644296420276801905236432, 4.20688340878182674339931011174, 4.74006328207835795732614245125, 5.62767145405384596348877496208, 6.50084583917956462186113244261, 6.83346147766309879590035918294, 7.897537864202885842321759356001, 8.656168874052238393920378245841