| L(s) = 1 | − 2-s + 4-s + 4·5-s − 8-s − 4·10-s − 11-s + 16-s − 17-s + 4·20-s + 22-s + 6·23-s + 11·25-s + 2·29-s − 4·31-s − 32-s + 34-s + 2·37-s − 4·40-s − 6·41-s − 4·43-s − 44-s − 6·46-s − 6·47-s − 11·50-s − 8·53-s − 4·55-s − 2·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.353·8-s − 1.26·10-s − 0.301·11-s + 1/4·16-s − 0.242·17-s + 0.894·20-s + 0.213·22-s + 1.25·23-s + 11/5·25-s + 0.371·29-s − 0.718·31-s − 0.176·32-s + 0.171·34-s + 0.328·37-s − 0.632·40-s − 0.937·41-s − 0.609·43-s − 0.150·44-s − 0.884·46-s − 0.875·47-s − 1.55·50-s − 1.09·53-s − 0.539·55-s − 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164934 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164934 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.32013327624216, −13.08035041160842, −12.72998193697423, −12.04881479857888, −11.31119445043715, −11.11544666740960, −10.41803590712730, −10.09926636896283, −9.706797415819950, −9.244060239367120, −8.746901738773306, −8.414758439537805, −7.722319511784961, −7.007106892456599, −6.725267854080220, −6.256836980693484, −5.604559314741705, −5.233568513438100, −4.784494084874393, −3.913607291873100, −2.976038837450412, −2.810329945888149, −1.985253693625534, −1.584600814750097, −0.9775965763608422, 0,
0.9775965763608422, 1.584600814750097, 1.985253693625534, 2.810329945888149, 2.976038837450412, 3.913607291873100, 4.784494084874393, 5.233568513438100, 5.604559314741705, 6.256836980693484, 6.725267854080220, 7.007106892456599, 7.722319511784961, 8.414758439537805, 8.746901738773306, 9.244060239367120, 9.706797415819950, 10.09926636896283, 10.41803590712730, 11.11544666740960, 11.31119445043715, 12.04881479857888, 12.72998193697423, 13.08035041160842, 13.32013327624216
