| L(s) = 1 | − 8·11-s − 8·13-s − 8·23-s − 2·25-s − 24·37-s − 24·47-s + 8·49-s − 24·59-s + 16·61-s + 16·71-s + 24·73-s − 9·81-s + 56·97-s + 32·107-s − 32·109-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 64·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 36·169-s + ⋯ |
| L(s) = 1 | − 2.41·11-s − 2.21·13-s − 1.66·23-s − 2/5·25-s − 3.94·37-s − 3.50·47-s + 8/7·49-s − 3.12·59-s + 2.04·61-s + 1.89·71-s + 2.80·73-s − 81-s + 5.68·97-s + 3.09·107-s − 3.06·109-s − 0.363·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.35·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.76·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4852849087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4852849087\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 18 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 13 | $D_{4}$ | \( ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 4 T + 44 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 470 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 + 36 T^{2} + 2150 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 7410 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 12 T + 124 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 11318 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 8 T + 114 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 9618 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 8 T + 134 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 28 T + 366 T^{2} - 28 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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
−6.59655441678853838811752721281, −6.42896644558877658119702306624, −6.17984692714758987043493697691, −5.82356876307410250197817252203, −5.53378606433946292123664802819, −5.31956343266321610278853432980, −5.31363040734891891155481386316, −5.17865964691239363458411811176, −4.77819087429970326287697439145, −4.69143699562721448389029417112, −4.65868721704601283607673988320, −4.04937795802564718577397406734, −4.03149006555780361561473303728, −3.47977715582030238510887554757, −3.40191639350684731660776267582, −3.17333351820532575230046618061, −3.04149343987728708260196947300, −2.63225489228734166587794567602, −2.12836691633548059577567627758, −2.12319551884840678185293684285, −1.93503638294745180975540520550, −1.88814021776235219400720709916, −1.09943476519666678636660774401, −0.40919772170541746698573003016, −0.22801809927001565572479610646,
0.22801809927001565572479610646, 0.40919772170541746698573003016, 1.09943476519666678636660774401, 1.88814021776235219400720709916, 1.93503638294745180975540520550, 2.12319551884840678185293684285, 2.12836691633548059577567627758, 2.63225489228734166587794567602, 3.04149343987728708260196947300, 3.17333351820532575230046618061, 3.40191639350684731660776267582, 3.47977715582030238510887554757, 4.03149006555780361561473303728, 4.04937795802564718577397406734, 4.65868721704601283607673988320, 4.69143699562721448389029417112, 4.77819087429970326287697439145, 5.17865964691239363458411811176, 5.31363040734891891155481386316, 5.31956343266321610278853432980, 5.53378606433946292123664802819, 5.82356876307410250197817252203, 6.17984692714758987043493697691, 6.42896644558877658119702306624, 6.59655441678853838811752721281
