| L(s) = 1 | + 2·3-s + 4-s + 9-s − 6·11-s + 2·12-s + 4·17-s + 12·19-s + 12·25-s − 2·27-s + 6·29-s − 12·33-s + 36-s − 6·37-s − 2·43-s − 6·44-s + 49-s + 8·51-s − 28·53-s + 24·57-s + 24·59-s + 20·61-s − 64-s − 24·67-s + 4·68-s + 18·71-s + 24·75-s + 12·76-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 1/3·9-s − 1.80·11-s + 0.577·12-s + 0.970·17-s + 2.75·19-s + 12/5·25-s − 0.384·27-s + 1.11·29-s − 2.08·33-s + 1/6·36-s − 0.986·37-s − 0.304·43-s − 0.904·44-s + 1/7·49-s + 1.12·51-s − 3.84·53-s + 3.17·57-s + 3.12·59-s + 2.56·61-s − 1/8·64-s − 2.93·67-s + 0.485·68-s + 2.13·71-s + 2.77·75-s + 1.37·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.987897485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.987897485\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 74 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 6 T + 3 p T^{2} + 126 T^{3} + 452 T^{4} + 126 p T^{5} + 3 p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T - 19 T^{2} - 4 T^{3} + 664 T^{4} - 4 p T^{5} - 19 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 12 T + 97 T^{2} - 588 T^{3} + 2952 T^{4} - 588 p T^{5} + 97 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 34 T^{2} + 627 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 28 T^{2} + 1818 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 6 T + 80 T^{2} + 408 T^{3} + 3699 T^{4} + 408 p T^{5} + 80 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 33 T^{2} - 592 T^{4} + 33 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 2 T - 80 T^{2} - 4 T^{3} + 5035 T^{4} - 4 p T^{5} - 80 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 162 T^{2} + 10931 T^{4} - 162 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 24 T + 322 T^{2} - 3120 T^{3} + 24747 T^{4} - 3120 p T^{5} + 322 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 20 T + 181 T^{2} - 1940 T^{3} + 19840 T^{4} - 1940 p T^{5} + 181 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 24 T + 338 T^{2} + 3504 T^{3} + 29691 T^{4} + 3504 p T^{5} + 338 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 18 T + 268 T^{2} - 2880 T^{3} + 28227 T^{4} - 2880 p T^{5} + 268 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 7014 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 17802 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T + 229 T^{2} - 2172 T^{3} + 29112 T^{4} - 2172 p T^{5} + 229 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 6 T + 208 T^{2} - 1176 T^{3} + 30171 T^{4} - 1176 p T^{5} + 208 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 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
−7.892806333294131667241497870368, −7.62487018256716504318075570008, −7.28455907419813849896668152094, −7.22502593975215821859943862949, −6.91614274089525594764423683270, −6.74340070011453172178382661878, −6.44132324408746688157840271153, −6.09624138311318828423350091838, −5.76025197894167119172042105702, −5.51732571543058982983138659094, −5.29322995276508959541377287710, −4.99532461089935335406458173485, −4.90708555079700418080452177215, −4.75952421466538603659828579620, −4.12979556107327993831553293039, −3.87355471689810194775368080787, −3.25466188175055117454751485025, −3.20087625567346718968660012328, −3.14365998135259039462833424574, −2.87647155089616607705287167428, −2.47473604089590103073627587386, −2.09372652888537309045567056225, −1.69309782111050767955684654096, −1.06637311802307695459857089553, −0.73515905067729712723373549967,
0.73515905067729712723373549967, 1.06637311802307695459857089553, 1.69309782111050767955684654096, 2.09372652888537309045567056225, 2.47473604089590103073627587386, 2.87647155089616607705287167428, 3.14365998135259039462833424574, 3.20087625567346718968660012328, 3.25466188175055117454751485025, 3.87355471689810194775368080787, 4.12979556107327993831553293039, 4.75952421466538603659828579620, 4.90708555079700418080452177215, 4.99532461089935335406458173485, 5.29322995276508959541377287710, 5.51732571543058982983138659094, 5.76025197894167119172042105702, 6.09624138311318828423350091838, 6.44132324408746688157840271153, 6.74340070011453172178382661878, 6.91614274089525594764423683270, 7.22502593975215821859943862949, 7.28455907419813849896668152094, 7.62487018256716504318075570008, 7.892806333294131667241497870368
