| L(s) = 1 | + 2-s − 9-s − 11-s − 18-s − 22-s + 25-s + 2·29-s − 32-s − 3·37-s + 50-s + 3·53-s + 2·58-s − 64-s − 5·71-s − 3·74-s + 5·79-s + 99-s + 3·106-s − 5·107-s + 2·109-s − 2·113-s + 127-s + 131-s + 137-s + 139-s − 5·142-s + 149-s + ⋯ |
| L(s) = 1 | + 2-s − 9-s − 11-s − 18-s − 22-s + 25-s + 2·29-s − 32-s − 3·37-s + 50-s + 3·53-s + 2·58-s − 64-s − 5·71-s − 3·74-s + 5·79-s + 99-s + 3·106-s − 5·107-s + 2·109-s − 2·113-s + 127-s + 131-s + 137-s + 139-s − 5·142-s + 149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.267134941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.267134941\) |
| \(L(1)\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 7 | | \( 1 \) |
| 11 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| good | 3 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 5 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 13 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 19 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 29 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 31 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 41 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 43 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 47 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 53 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 61 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 67 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 71 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 79 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 83 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.59790669053554305356205092444, −6.52730173290851273359128547822, −6.25604237199220609150750650274, −5.88619258511561184115702576009, −5.60628586288114440575739415966, −5.55000621923335633472208920160, −5.39930400202314447065443570226, −5.20392382048623533902776014333, −4.83291672661931920139357735444, −4.80219421442290289189964615938, −4.78564509680483174264859772263, −4.25746221996550414727774721627, −3.99432370900219462247714018816, −3.92023985262677330884460178430, −3.75375999522768963027245614725, −3.29622688538725096910354894141, −3.02559129580826860242237050292, −3.00846288659231015062940276482, −2.71655429696924485186212542077, −2.32198143447043050283840273335, −2.29961280087197328923065562649, −1.77272992327997460524891031736, −1.44882292241068811408419734826, −1.12812718897461128216231276572, −0.47548975821345062308886031626,
0.47548975821345062308886031626, 1.12812718897461128216231276572, 1.44882292241068811408419734826, 1.77272992327997460524891031736, 2.29961280087197328923065562649, 2.32198143447043050283840273335, 2.71655429696924485186212542077, 3.00846288659231015062940276482, 3.02559129580826860242237050292, 3.29622688538725096910354894141, 3.75375999522768963027245614725, 3.92023985262677330884460178430, 3.99432370900219462247714018816, 4.25746221996550414727774721627, 4.78564509680483174264859772263, 4.80219421442290289189964615938, 4.83291672661931920139357735444, 5.20392382048623533902776014333, 5.39930400202314447065443570226, 5.55000621923335633472208920160, 5.60628586288114440575739415966, 5.88619258511561184115702576009, 6.25604237199220609150750650274, 6.52730173290851273359128547822, 6.59790669053554305356205092444
