L(s) = 1 | − 2-s − 3·3-s + 4-s + 2·5-s + 3·6-s − 3·7-s − 8-s + 6·9-s − 2·10-s − 2·11-s − 3·12-s + 3·13-s + 3·14-s − 6·15-s + 16-s − 17-s − 6·18-s + 2·20-s + 9·21-s + 2·22-s + 5·23-s + 3·24-s − 25-s − 3·26-s − 9·27-s − 3·28-s + 3·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.894·5-s + 1.22·6-s − 1.13·7-s − 0.353·8-s + 2·9-s − 0.632·10-s − 0.603·11-s − 0.866·12-s + 0.832·13-s + 0.801·14-s − 1.54·15-s + 1/4·16-s − 0.242·17-s − 1.41·18-s + 0.447·20-s + 1.96·21-s + 0.426·22-s + 1.04·23-s + 0.612·24-s − 1/5·25-s − 0.588·26-s − 1.73·27-s − 0.566·28-s + 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−10.24841120930238485398404246571, −9.440803609961611917450717039072, −8.359856169166879769635379775286, −6.86815419256923705153210540276, −6.50910383072950624405233258608, −5.72634740725250578850981545629, −4.87044643269311183195194730344, −3.17689138557574316956585675012, −1.48667914659191059870015189789, 0,
1.48667914659191059870015189789, 3.17689138557574316956585675012, 4.87044643269311183195194730344, 5.72634740725250578850981545629, 6.50910383072950624405233258608, 6.86815419256923705153210540276, 8.359856169166879769635379775286, 9.440803609961611917450717039072, 10.24841120930238485398404246571