| L(s) = 1 | + 3-s + 4·7-s + 9-s + 4·11-s − 2·17-s − 4·19-s + 4·21-s − 8·23-s − 5·25-s + 27-s + 8·29-s + 4·31-s + 4·33-s − 4·37-s − 6·41-s − 4·43-s + 8·47-s + 9·49-s − 2·51-s + 8·53-s − 4·57-s − 12·59-s − 12·61-s + 4·63-s + 12·67-s − 8·69-s − 8·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.485·17-s − 0.917·19-s + 0.872·21-s − 1.66·23-s − 25-s + 0.192·27-s + 1.48·29-s + 0.718·31-s + 0.696·33-s − 0.657·37-s − 0.937·41-s − 0.609·43-s + 1.16·47-s + 9/7·49-s − 0.280·51-s + 1.09·53-s − 0.529·57-s − 1.56·59-s − 1.53·61-s + 0.503·63-s + 1.46·67-s − 0.963·69-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.84632485852423, −13.61301779892043, −12.65612692563275, −12.14588716422748, −11.81686488794020, −11.50375867618190, −10.78091479040594, −10.26469460774519, −10.01026634163940, −9.119316660321574, −8.835030303461409, −8.301993906407356, −7.994835599419489, −7.472664399012031, −6.752170824522654, −6.319364319212218, −5.836082921793004, −4.948405853391965, −4.587658892510076, −4.018137980389590, −3.717734910052536, −2.725259530197182, −2.108553518802638, −1.682452792634301, −1.108458439911132, 0,
1.108458439911132, 1.682452792634301, 2.108553518802638, 2.725259530197182, 3.717734910052536, 4.018137980389590, 4.587658892510076, 4.948405853391965, 5.836082921793004, 6.319364319212218, 6.752170824522654, 7.472664399012031, 7.994835599419489, 8.301993906407356, 8.835030303461409, 9.119316660321574, 10.01026634163940, 10.26469460774519, 10.78091479040594, 11.50375867618190, 11.81686488794020, 12.14588716422748, 12.65612692563275, 13.61301779892043, 13.84632485852423
