L(s) = 1 | + 2-s − 4-s − 5-s + 2·7-s − 3·8-s − 10-s − 13-s + 2·14-s − 16-s − 5·17-s − 6·19-s + 20-s − 2·23-s − 4·25-s − 26-s − 2·28-s + 9·29-s − 2·31-s + 5·32-s − 5·34-s − 2·35-s − 3·37-s − 6·38-s + 3·40-s − 5·41-s − 2·46-s − 2·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s + 0.755·7-s − 1.06·8-s − 0.316·10-s − 0.277·13-s + 0.534·14-s − 1/4·16-s − 1.21·17-s − 1.37·19-s + 0.223·20-s − 0.417·23-s − 4/5·25-s − 0.196·26-s − 0.377·28-s + 1.67·29-s − 0.359·31-s + 0.883·32-s − 0.857·34-s − 0.338·35-s − 0.493·37-s − 0.973·38-s + 0.474·40-s − 0.780·41-s − 0.294·46-s − 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−9.346568315137948705424383943018, −8.507285126107606377633981082529, −8.010979443598278861073897866124, −6.74272981252089439582006678154, −5.97336865191641765106303088244, −4.68607726153727393582211790246, −4.50122243823954701224397495329, −3.33556810952068997685484956976, −2.02313919782203708817074154348, 0,
2.02313919782203708817074154348, 3.33556810952068997685484956976, 4.50122243823954701224397495329, 4.68607726153727393582211790246, 5.97336865191641765106303088244, 6.74272981252089439582006678154, 8.010979443598278861073897866124, 8.507285126107606377633981082529, 9.346568315137948705424383943018