L(s) = 1 | − 2·2-s + 2·4-s + 5-s − 2·10-s − 4·11-s + 2·13-s − 4·16-s − 3·17-s + 8·19-s + 2·20-s + 8·22-s − 6·23-s − 4·25-s − 4·26-s − 4·29-s − 6·31-s + 8·32-s + 6·34-s − 3·37-s − 16·38-s − 41-s + 11·43-s − 8·44-s + 12·46-s − 9·47-s + 8·50-s + 4·52-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s + 0.447·5-s − 0.632·10-s − 1.20·11-s + 0.554·13-s − 16-s − 0.727·17-s + 1.83·19-s + 0.447·20-s + 1.70·22-s − 1.25·23-s − 4/5·25-s − 0.784·26-s − 0.742·29-s − 1.07·31-s + 1.41·32-s + 1.02·34-s − 0.493·37-s − 2.59·38-s − 0.156·41-s + 1.67·43-s − 1.20·44-s + 1.76·46-s − 1.31·47-s + 1.13·50-s + 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−9.310424015638182849224826873304, −8.497771804391192154093576901064, −7.69931536429066472922525605470, −7.21149427744505179683222147621, −5.96598102544836283817916841913, −5.23983624007258288531004002941, −3.90524998328143242553926635429, −2.52420740459652612310538151640, −1.53430709834964791772483315213, 0,
1.53430709834964791772483315213, 2.52420740459652612310538151640, 3.90524998328143242553926635429, 5.23983624007258288531004002941, 5.96598102544836283817916841913, 7.21149427744505179683222147621, 7.69931536429066472922525605470, 8.497771804391192154093576901064, 9.310424015638182849224826873304