L(s) = 1 | − 2-s + 4-s − 2·5-s + 7-s − 8-s + 2·10-s + 4·13-s − 14-s + 16-s − 17-s − 4·19-s − 2·20-s + 8·23-s − 25-s − 4·26-s + 28-s + 8·29-s − 4·31-s − 32-s + 34-s − 2·35-s + 6·37-s + 4·38-s + 2·40-s − 6·41-s − 2·43-s − 8·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s − 0.353·8-s + 0.632·10-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.917·19-s − 0.447·20-s + 1.66·23-s − 1/5·25-s − 0.784·26-s + 0.188·28-s + 1.48·29-s − 0.718·31-s − 0.176·32-s + 0.171·34-s − 0.338·35-s + 0.986·37-s + 0.648·38-s + 0.316·40-s − 0.937·41-s − 0.304·43-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−12.96288815608321, −12.60408623320482, −11.85140546584353, −11.65963871344174, −11.06480256962807, −10.87056800448334, −10.40294216891527, −9.778741859932121, −9.232219383414268, −8.735691468326171, −8.408819027004096, −7.999449166160492, −7.599173975666767, −6.901147686396039, −6.495418811756514, −6.231193458244456, −5.279756483928833, −4.933026804576104, −4.309292752615603, −3.718642361007202, −3.304691992328280, −2.648101652693151, −1.984365459701136, −1.296414104640936, −0.7754723439433850, 0,
0.7754723439433850, 1.296414104640936, 1.984365459701136, 2.648101652693151, 3.304691992328280, 3.718642361007202, 4.309292752615603, 4.933026804576104, 5.279756483928833, 6.231193458244456, 6.495418811756514, 6.901147686396039, 7.599173975666767, 7.999449166160492, 8.408819027004096, 8.735691468326171, 9.232219383414268, 9.778741859932121, 10.40294216891527, 10.87056800448334, 11.06480256962807, 11.65963871344174, 11.85140546584353, 12.60408623320482, 12.96288815608321