| L(s) = 1 | − 2·3-s − 7-s + 9-s − 4·13-s − 6·17-s − 2·19-s + 2·21-s + 4·27-s + 6·29-s − 4·31-s + 2·37-s + 8·39-s + 6·41-s + 8·43-s + 12·47-s + 49-s + 12·51-s + 6·53-s + 4·57-s + 6·59-s − 8·61-s − 63-s − 4·67-s − 2·73-s + 8·79-s − 11·81-s − 6·83-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s − 1.10·13-s − 1.45·17-s − 0.458·19-s + 0.436·21-s + 0.769·27-s + 1.11·29-s − 0.718·31-s + 0.328·37-s + 1.28·39-s + 0.937·41-s + 1.21·43-s + 1.75·47-s + 1/7·49-s + 1.68·51-s + 0.824·53-s + 0.529·57-s + 0.781·59-s − 1.02·61-s − 0.125·63-s − 0.488·67-s − 0.234·73-s + 0.900·79-s − 1.22·81-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| 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
−16.96997140511505, −16.23946831441621, −15.72233842021987, −15.21705667587369, −14.45769853004689, −13.95298425575299, −13.15181859447658, −12.61679281065683, −12.16443569526269, −11.60279925785864, −10.90709848788717, −10.57090429093518, −9.910129911569018, −9.064899952938122, −8.736817921425390, −7.646737493402776, −7.116426497427377, −6.445992450933586, −5.949660329000150, −5.252361962889186, −4.539244496369989, −4.052507081272153, −2.757917478045365, −2.253260617155550, −0.8562256092823503, 0,
0.8562256092823503, 2.253260617155550, 2.757917478045365, 4.052507081272153, 4.539244496369989, 5.252361962889186, 5.949660329000150, 6.445992450933586, 7.116426497427377, 7.646737493402776, 8.736817921425390, 9.064899952938122, 9.910129911569018, 10.57090429093518, 10.90709848788717, 11.60279925785864, 12.16443569526269, 12.61679281065683, 13.15181859447658, 13.95298425575299, 14.45769853004689, 15.21705667587369, 15.72233842021987, 16.23946831441621, 16.96997140511505
