| L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s + 7-s + 3·8-s + 9-s + 10-s − 2·11-s + 12-s − 14-s + 15-s − 16-s + 17-s − 18-s + 8·19-s + 20-s − 21-s + 2·22-s − 6·23-s − 3·24-s + 25-s − 27-s − 28-s + 6·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s − 0.267·14-s + 0.258·15-s − 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.83·19-s + 0.223·20-s − 0.218·21-s + 0.426·22-s − 1.25·23-s − 0.612·24-s + 1/5·25-s − 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−12.86984059024108, −12.25039588747053, −12.00228624212366, −11.53492895098163, −10.96328041016532, −10.63855842896155, −10.00390688566191, −9.829832936156340, −9.342822311283732, −8.696561096794310, −8.212460795656522, −7.813359843694712, −7.604563803234612, −6.942765734767493, −6.423499861596750, −5.760487527416883, −5.216250612784867, −4.867780887473434, −4.518210447756476, −3.660654393240344, −3.406780479536000, −2.589198077306340, −1.758525312279642, −1.279293111732775, −0.6290014641904774, 0,
0.6290014641904774, 1.279293111732775, 1.758525312279642, 2.589198077306340, 3.406780479536000, 3.660654393240344, 4.518210447756476, 4.867780887473434, 5.216250612784867, 5.760487527416883, 6.423499861596750, 6.942765734767493, 7.604563803234612, 7.813359843694712, 8.212460795656522, 8.696561096794310, 9.342822311283732, 9.829832936156340, 10.00390688566191, 10.63855842896155, 10.96328041016532, 11.53492895098163, 12.00228624212366, 12.25039588747053, 12.86984059024108
