L(s) = 1 | − 2-s − 2·3-s + 2·6-s + 9-s − 11-s − 2·17-s − 18-s + 3·19-s + 22-s − 25-s + 32-s + 2·33-s + 2·34-s − 3·38-s − 2·41-s − 2·43-s − 49-s + 50-s + 4·51-s − 6·57-s + 3·59-s − 64-s − 2·66-s − 2·67-s − 2·73-s + 2·75-s + 2·82-s + ⋯ |
L(s) = 1 | − 2-s − 2·3-s + 2·6-s + 9-s − 11-s − 2·17-s − 18-s + 3·19-s + 22-s − 25-s + 32-s + 2·33-s + 2·34-s − 3·38-s − 2·41-s − 2·43-s − 49-s + 50-s + 4·51-s − 6·57-s + 3·59-s − 64-s − 2·66-s − 2·67-s − 2·73-s + 2·75-s + 2·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59969536 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59969536 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03037514870\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03037514870\) |
\(L(1)\) |
|
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
good | 3 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 5 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 7 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 13 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 31 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 41 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 43 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 47 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 53 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 61 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 67 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 97 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.07767912405533495553158013088, −10.25751161800844311628585574119, −10.15619435925391413893199930499, −10.13125239066839070428183229427, −9.801542251643198264879174078581, −9.284969942379507859608611312768, −9.191973812477788515768121620020, −8.700097417800058409783457980085, −8.433898624247411372751716098663, −8.341090551312655104382721172892, −7.66367238784448783056406444752, −7.48952021262817008111292203391, −7.28039020908791378434224755870, −6.66545754783887735413245583256, −6.38456810366459715900409676471, −6.29990722230209404329316790795, −5.50038095571044670886224694877, −5.47635468920967000080472360694, −5.32559889154914950171752856996, −4.65735988994615014311933168328, −4.63376475369904117861461445186, −3.67884878259526706903463378333, −3.29597448370317997586963941688, −2.68791914585162550379183914391, −1.80284319275365030447112589300,
1.80284319275365030447112589300, 2.68791914585162550379183914391, 3.29597448370317997586963941688, 3.67884878259526706903463378333, 4.63376475369904117861461445186, 4.65735988994615014311933168328, 5.32559889154914950171752856996, 5.47635468920967000080472360694, 5.50038095571044670886224694877, 6.29990722230209404329316790795, 6.38456810366459715900409676471, 6.66545754783887735413245583256, 7.28039020908791378434224755870, 7.48952021262817008111292203391, 7.66367238784448783056406444752, 8.341090551312655104382721172892, 8.433898624247411372751716098663, 8.700097417800058409783457980085, 9.191973812477788515768121620020, 9.284969942379507859608611312768, 9.801542251643198264879174078581, 10.13125239066839070428183229427, 10.15619435925391413893199930499, 10.25751161800844311628585574119, 11.07767912405533495553158013088