L(s) = 1 | − 2·2-s + 2·4-s − 3·5-s − 7-s − 3·9-s + 6·10-s − 6·11-s − 13-s + 2·14-s − 4·16-s + 4·17-s + 6·18-s + 5·19-s − 6·20-s + 12·22-s + 3·23-s + 4·25-s + 2·26-s − 2·28-s − 5·29-s − 3·31-s + 8·32-s − 8·34-s + 3·35-s − 6·36-s − 4·37-s − 10·38-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s − 1.34·5-s − 0.377·7-s − 9-s + 1.89·10-s − 1.80·11-s − 0.277·13-s + 0.534·14-s − 16-s + 0.970·17-s + 1.41·18-s + 1.14·19-s − 1.34·20-s + 2.55·22-s + 0.625·23-s + 4/5·25-s + 0.392·26-s − 0.377·28-s − 0.928·29-s − 0.538·31-s + 1.41·32-s − 1.37·34-s + 0.507·35-s − 36-s − 0.657·37-s − 1.62·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.50802048420851007566361386612, −12.13349961472521897100566393596, −11.15186582749693924229422906687, −10.25104185337052377091134397992, −8.998346772410808154035420365472, −7.88305646625584781075065151406, −7.43563929744309328324545511818, −5.27883313770925671699270459709, −3.11744276397134147631028514628, 0,
3.11744276397134147631028514628, 5.27883313770925671699270459709, 7.43563929744309328324545511818, 7.88305646625584781075065151406, 8.998346772410808154035420365472, 10.25104185337052377091134397992, 11.15186582749693924229422906687, 12.13349961472521897100566393596, 13.50802048420851007566361386612