L(s) = 1 | − 2·4-s + 5-s − 4·7-s + 13-s + 4·16-s − 3·17-s + 2·19-s − 2·20-s + 3·23-s + 25-s + 8·28-s + 6·29-s − 4·31-s − 4·35-s + 8·37-s − 12·41-s − 43-s + 9·49-s − 2·52-s − 3·53-s + 6·59-s + 11·61-s − 8·64-s + 65-s − 4·67-s + 6·68-s + 6·71-s + ⋯ |
L(s) = 1 | − 4-s + 0.447·5-s − 1.51·7-s + 0.277·13-s + 16-s − 0.727·17-s + 0.458·19-s − 0.447·20-s + 0.625·23-s + 1/5·25-s + 1.51·28-s + 1.11·29-s − 0.718·31-s − 0.676·35-s + 1.31·37-s − 1.87·41-s − 0.152·43-s + 9/7·49-s − 0.277·52-s − 0.412·53-s + 0.781·59-s + 1.40·61-s − 64-s + 0.124·65-s − 0.488·67-s + 0.727·68-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.994495867514867482794428440919, −6.87559778169481118278634407670, −6.49943559453733683768346384949, −5.61280054436949515473753268234, −4.98067605090956134286853028451, −4.03682379199348071482918145285, −3.34911405005937571002812786699, −2.56298718065653629459640519266, −1.12581694770082408676813926761, 0,
1.12581694770082408676813926761, 2.56298718065653629459640519266, 3.34911405005937571002812786699, 4.03682379199348071482918145285, 4.98067605090956134286853028451, 5.61280054436949515473753268234, 6.49943559453733683768346384949, 6.87559778169481118278634407670, 7.994495867514867482794428440919