| L(s) = 1 | + 2·13-s + 2·19-s − 6·29-s + 8·31-s + 4·37-s + 6·41-s + 2·43-s − 6·47-s + 6·53-s − 12·59-s − 8·61-s + 2·67-s + 6·71-s + 2·73-s + 16·79-s + 6·89-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.554·13-s + 0.458·19-s − 1.11·29-s + 1.43·31-s + 0.657·37-s + 0.937·41-s + 0.304·43-s − 0.875·47-s + 0.824·53-s − 1.56·59-s − 1.02·61-s + 0.244·67-s + 0.712·71-s + 0.234·73-s + 1.80·79-s + 0.635·89-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.49578112448531, −12.94088982829378, −12.48002584536418, −12.02823824686007, −11.47202202276700, −11.08780491198522, −10.66186746162246, −10.08325494998839, −9.554491551474145, −9.203145114632914, −8.686361383010448, −8.006856129215390, −7.753641133008332, −7.210395703236182, −6.475459964170059, −6.171920924210468, −5.664534271363509, −4.949553274381565, −4.604280874234235, −3.825077052503235, −3.498599217437739, −2.718719589028006, −2.270954061144040, −1.400638559758660, −0.9208817541614280, 0,
0.9208817541614280, 1.400638559758660, 2.270954061144040, 2.718719589028006, 3.498599217437739, 3.825077052503235, 4.604280874234235, 4.949553274381565, 5.664534271363509, 6.171920924210468, 6.475459964170059, 7.210395703236182, 7.753641133008332, 8.006856129215390, 8.686361383010448, 9.203145114632914, 9.554491551474145, 10.08325494998839, 10.66186746162246, 11.08780491198522, 11.47202202276700, 12.02823824686007, 12.48002584536418, 12.94088982829378, 13.49578112448531
