L(s) = 1 | + 2-s − 4-s − 3·8-s − 5·11-s − 5·13-s − 16-s + 4·17-s − 2·19-s − 5·22-s − 3·23-s − 5·26-s − 10·29-s + 6·31-s + 5·32-s + 4·34-s + 5·37-s − 2·38-s − 10·41-s + 10·43-s + 5·44-s − 3·46-s − 5·47-s − 7·49-s + 5·52-s − 2·53-s − 10·58-s − 5·59-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 1.50·11-s − 1.38·13-s − 1/4·16-s + 0.970·17-s − 0.458·19-s − 1.06·22-s − 0.625·23-s − 0.980·26-s − 1.85·29-s + 1.07·31-s + 0.883·32-s + 0.685·34-s + 0.821·37-s − 0.324·38-s − 1.56·41-s + 1.52·43-s + 0.753·44-s − 0.442·46-s − 0.729·47-s − 49-s + 0.693·52-s − 0.274·53-s − 1.31·58-s − 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−9.946684219512384164367328331430, −9.432452334454903655823372500069, −8.111291554543981630243615144391, −7.61064821647543776585659150038, −6.20481852631863748855544323115, −5.27951774771441848148740620672, −4.69824702877576535283262083774, −3.45045562084777234969796191815, −2.40058573044279568120832059087, 0,
2.40058573044279568120832059087, 3.45045562084777234969796191815, 4.69824702877576535283262083774, 5.27951774771441848148740620672, 6.20481852631863748855544323115, 7.61064821647543776585659150038, 8.111291554543981630243615144391, 9.432452334454903655823372500069, 9.946684219512384164367328331430