| L(s) = 1 | − 2·3-s + 4-s − 8·5-s − 3·9-s + 2·11-s − 2·12-s + 16·15-s + 16-s − 8·20-s − 2·23-s + 38·25-s + 14·27-s − 16·31-s − 4·33-s − 3·36-s − 4·37-s + 2·44-s + 24·45-s + 16·47-s − 2·48-s − 5·49-s − 2·53-s − 16·55-s + 30·59-s + 16·60-s + 64-s + 6·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s − 3.57·5-s − 9-s + 0.603·11-s − 0.577·12-s + 4.13·15-s + 1/4·16-s − 1.78·20-s − 0.417·23-s + 38/5·25-s + 2.69·27-s − 2.87·31-s − 0.696·33-s − 1/2·36-s − 0.657·37-s + 0.301·44-s + 3.57·45-s + 2.33·47-s − 0.288·48-s − 5/7·49-s − 0.274·53-s − 2.15·55-s + 3.90·59-s + 2.06·60-s + 1/8·64-s + 0.733·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.713028473107534219835074143035, −8.408486909011940238981299566497, −7.979676687352676305303385977682, −7.25038064772684706053554512263, −7.24993124552845565031332433274, −6.68239009544393402711079299710, −5.98269892429734678737463344232, −5.28069074518443602737935578596, −5.02409024164795186610457056564, −3.99098546974644050534648424657, −3.86057015395193111701613139713, −3.39047289495238098925090461140, −2.52867986473958162980513935632, −0.76546142649705169453989550316, 0,
0.76546142649705169453989550316, 2.52867986473958162980513935632, 3.39047289495238098925090461140, 3.86057015395193111701613139713, 3.99098546974644050534648424657, 5.02409024164795186610457056564, 5.28069074518443602737935578596, 5.98269892429734678737463344232, 6.68239009544393402711079299710, 7.24993124552845565031332433274, 7.25038064772684706053554512263, 7.979676687352676305303385977682, 8.408486909011940238981299566497, 8.713028473107534219835074143035
