| L(s) = 1 | − 2-s + 4-s − 5-s − 4·7-s − 8-s + 10-s − 2·13-s + 4·14-s + 16-s − 2·17-s − 4·19-s − 20-s + 25-s + 2·26-s − 4·28-s + 2·29-s − 32-s + 2·34-s + 4·35-s − 10·37-s + 4·38-s + 40-s + 6·41-s + 8·47-s + 9·49-s − 50-s − 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s − 0.353·8-s + 0.316·10-s − 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.223·20-s + 1/5·25-s + 0.392·26-s − 0.755·28-s + 0.371·29-s − 0.176·32-s + 0.342·34-s + 0.676·35-s − 1.64·37-s + 0.648·38-s + 0.158·40-s + 0.937·41-s + 1.16·47-s + 9/7·49-s − 0.141·50-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 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
−13.27557170358466, −12.94146806504353, −12.43417394001812, −12.14373269206351, −11.59470017285475, −10.91944421180896, −10.60290766266265, −10.04353057900779, −9.755486385606658, −9.086064737683382, −8.740645633668316, −8.391067643881657, −7.447558537131477, −7.328368440921139, −6.725684063031208, −6.253521704064655, −5.855382815408408, −5.097240094228857, −4.415332343354833, −3.891836083234793, −3.303429306918961, −2.726684118069771, −2.275689451281589, −1.447890181973997, −0.5197558704223796, 0,
0.5197558704223796, 1.447890181973997, 2.275689451281589, 2.726684118069771, 3.303429306918961, 3.891836083234793, 4.415332343354833, 5.097240094228857, 5.855382815408408, 6.253521704064655, 6.725684063031208, 7.328368440921139, 7.447558537131477, 8.391067643881657, 8.740645633668316, 9.086064737683382, 9.755486385606658, 10.04353057900779, 10.60290766266265, 10.91944421180896, 11.59470017285475, 12.14373269206351, 12.43417394001812, 12.94146806504353, 13.27557170358466
