| L(s) = 1 | − 3·3-s + 2·5-s + 6·9-s − 5·11-s − 7·13-s − 6·15-s + 17-s + 2·19-s − 25-s − 9·27-s − 6·29-s + 4·31-s + 15·33-s + 8·37-s + 21·39-s − 2·41-s − 8·43-s + 12·45-s − 10·47-s − 3·51-s + 3·53-s − 10·55-s − 6·57-s + 12·61-s − 14·65-s − 2·67-s + 71-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.894·5-s + 2·9-s − 1.50·11-s − 1.94·13-s − 1.54·15-s + 0.242·17-s + 0.458·19-s − 1/5·25-s − 1.73·27-s − 1.11·29-s + 0.718·31-s + 2.61·33-s + 1.31·37-s + 3.36·39-s − 0.312·41-s − 1.21·43-s + 1.78·45-s − 1.45·47-s − 0.420·51-s + 0.412·53-s − 1.34·55-s − 0.794·57-s + 1.53·61-s − 1.73·65-s − 0.244·67-s + 0.118·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−14.90693002037730, −14.15512766665973, −13.42326018600698, −13.02464011200494, −12.75730631562612, −11.93306670144574, −11.79348427984423, −11.12514740857941, −10.54521252692900, −10.07197275912590, −9.750099728572547, −9.443566154161585, −8.242808192136098, −7.727835542489029, −7.251201482393660, −6.636694826422900, −6.106275877905619, −5.468845146562127, −5.062986234600519, −4.945625024441987, −4.025758747371650, −2.983376990762342, −2.321590907095338, −1.728887311428013, −0.6715797427515093, 0,
0.6715797427515093, 1.728887311428013, 2.321590907095338, 2.983376990762342, 4.025758747371650, 4.945625024441987, 5.062986234600519, 5.468845146562127, 6.106275877905619, 6.636694826422900, 7.251201482393660, 7.727835542489029, 8.242808192136098, 9.443566154161585, 9.750099728572547, 10.07197275912590, 10.54521252692900, 11.12514740857941, 11.79348427984423, 11.93306670144574, 12.75730631562612, 13.02464011200494, 13.42326018600698, 14.15512766665973, 14.90693002037730
