| 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 + 5·19-s − 20-s + 21-s − 2·22-s − 3·23-s − 3·24-s + 25-s + 27-s − 28-s − 5·29-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 + 1.14·19-s − 0.223·20-s + 0.218·21-s − 0.426·22-s − 0.625·23-s − 0.612·24-s + 1/5·25-s + 0.192·27-s − 0.188·28-s − 0.928·29-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 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.95718974009267, −12.63325373374788, −12.11196479164568, −11.71835060433953, −11.07646690713348, −10.64993968848645, −10.05995802933586, −9.570087145875758, −9.348190581391947, −8.780631699104218, −8.278135286483995, −7.890901684315442, −7.304736761645077, −6.904853853685569, −6.187610964425844, −5.565698254105999, −5.268694787992919, −5.035256493097351, −4.184135174298156, −3.737781536804956, −3.364062350839488, −2.786952749076249, −2.084502695755034, −1.678675560635597, −0.7999769925560096, 0,
0.7999769925560096, 1.678675560635597, 2.084502695755034, 2.786952749076249, 3.364062350839488, 3.737781536804956, 4.184135174298156, 5.035256493097351, 5.268694787992919, 5.565698254105999, 6.187610964425844, 6.904853853685569, 7.304736761645077, 7.890901684315442, 8.278135286483995, 8.780631699104218, 9.348190581391947, 9.570087145875758, 10.05995802933586, 10.64993968848645, 11.07646690713348, 11.71835060433953, 12.11196479164568, 12.63325373374788, 12.95718974009267
