| L(s) = 1 | + 2-s − 4-s − 2·5-s − 3·8-s − 2·10-s − 2·13-s − 16-s + 2·17-s + 2·20-s − 8·23-s − 25-s − 2·26-s − 6·29-s + 8·31-s + 5·32-s + 2·34-s + 6·37-s + 6·40-s + 2·41-s − 8·46-s + 8·47-s − 50-s + 2·52-s − 6·53-s − 6·58-s − 4·59-s + 6·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.06·8-s − 0.632·10-s − 0.554·13-s − 1/4·16-s + 0.485·17-s + 0.447·20-s − 1.66·23-s − 1/5·25-s − 0.392·26-s − 1.11·29-s + 1.43·31-s + 0.883·32-s + 0.342·34-s + 0.986·37-s + 0.948·40-s + 0.312·41-s − 1.17·46-s + 1.16·47-s − 0.141·50-s + 0.277·52-s − 0.824·53-s − 0.787·58-s − 0.520·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7223344678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7223344678\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−14.42857663342633, −13.91538684077652, −13.52156375397412, −12.90435102303871, −12.27154831047292, −12.13711372554740, −11.53908250645419, −11.08792775390205, −10.18774199202446, −9.817909748889228, −9.357047312138147, −8.575125485858346, −8.157303165849052, −7.625897740248053, −7.185163660926686, −6.137446966621650, −5.973497764684955, −5.234419549490064, −4.553235335170780, −4.106085229356769, −3.740959696628894, −2.932276251029419, −2.380673740129956, −1.293068284211579, −0.2758002527941792,
0.2758002527941792, 1.293068284211579, 2.380673740129956, 2.932276251029419, 3.740959696628894, 4.106085229356769, 4.553235335170780, 5.234419549490064, 5.973497764684955, 6.137446966621650, 7.185163660926686, 7.625897740248053, 8.157303165849052, 8.575125485858346, 9.357047312138147, 9.817909748889228, 10.18774199202446, 11.08792775390205, 11.53908250645419, 12.13711372554740, 12.27154831047292, 12.90435102303871, 13.52156375397412, 13.91538684077652, 14.42857663342633
