| L(s) = 1 | − 2-s − 4-s + 5-s + 2·7-s + 3·8-s − 10-s + 4·11-s − 2·13-s − 2·14-s − 16-s − 6·17-s − 20-s − 4·22-s − 6·23-s + 25-s + 2·26-s − 2·28-s − 4·31-s − 5·32-s + 6·34-s + 2·35-s + 6·37-s + 3·40-s + 8·41-s + 6·43-s − 4·44-s + 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 0.755·7-s + 1.06·8-s − 0.316·10-s + 1.20·11-s − 0.554·13-s − 0.534·14-s − 1/4·16-s − 1.45·17-s − 0.223·20-s − 0.852·22-s − 1.25·23-s + 1/5·25-s + 0.392·26-s − 0.377·28-s − 0.718·31-s − 0.883·32-s + 1.02·34-s + 0.338·35-s + 0.986·37-s + 0.474·40-s + 1.24·41-s + 0.914·43-s − 0.603·44-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−16.26273937756855, −15.91727906104058, −14.88760424800857, −14.57938440149819, −13.94184778502602, −13.72267683213131, −12.70194159591985, −12.54615258919076, −11.46095993136365, −11.16924453010695, −10.58888832245443, −9.677299850500546, −9.476642633847634, −8.933325272809587, −8.244521579974750, −7.792224544737773, −7.044489712575237, −6.396132570885751, −5.694102927777490, −4.872375169060376, −4.286782733654280, −3.875108661727217, −2.508016437010309, −1.849992287269097, −1.117190259165496, 0,
1.117190259165496, 1.849992287269097, 2.508016437010309, 3.875108661727217, 4.286782733654280, 4.872375169060376, 5.694102927777490, 6.396132570885751, 7.044489712575237, 7.792224544737773, 8.244521579974750, 8.933325272809587, 9.476642633847634, 9.677299850500546, 10.58888832245443, 11.16924453010695, 11.46095993136365, 12.54615258919076, 12.70194159591985, 13.72267683213131, 13.94184778502602, 14.57938440149819, 14.88760424800857, 15.91727906104058, 16.26273937756855
