| L(s) = 1 | − 2·16-s + 24·29-s − 16·31-s − 24·59-s − 32·61-s − 24·89-s + 52·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | − 1/2·16-s + 4.45·29-s − 2.87·31-s − 3.12·59-s − 4.09·61-s − 2.54·89-s + 4.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02372807700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02372807700\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4 p T^{4} + 3270 T^{8} + 4 p^{5} T^{12} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 49 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 + 796 T^{4} + 309894 T^{8} + 796 p^{4} T^{12} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 20 T^{2} + 714 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 311 T^{4} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 37 | \( 1 - 1442 T^{4} + 2270595 T^{8} - 1442 p^{4} T^{12} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 - 5444 T^{4} + 14231334 T^{8} - 5444 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 4249 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 + 508 T^{4} - 3205722 T^{8} + 508 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 8 T + 111 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( 1 + 3196 T^{4} + 5515494 T^{8} + 3196 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 158 T^{2} + 12435 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 - 7292 T^{4} + 67324998 T^{8} - 7292 p^{4} T^{12} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 212 T^{2} + 21018 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 4516 T^{4} + 69906534 T^{8} + 4516 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 6 T + 160 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 - 2258 T^{4} - 119271597 T^{8} - 2258 p^{4} T^{12} + p^{8} T^{16} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.94279230562842542456720748285, −3.87864838548261594036174125617, −3.74299674049081273310336159002, −3.66552266549810793932031855570, −3.65367636488729184176222526808, −3.59047065985743754673622987913, −3.22356086965437554225874724068, −3.11616100188518883379337162345, −2.90418213591286321479388772324, −2.90404489858447869689746762304, −2.82244357483652658911517576183, −2.80367136786194763605564137039, −2.41718873513721421987817455832, −2.38587209817188255723455286720, −2.22366744686821132209482183953, −2.03879200196406574979034740172, −1.70118809959118278879721452230, −1.68997328924518543236132507240, −1.62247224845569329710186986049, −1.23168300996773421820606583965, −1.20506266560044681773564449660, −1.03215239411056794736302919822, −0.846012620433440372408705899671, −0.27501041165787316827732023863, −0.02504608039845248947860412154,
0.02504608039845248947860412154, 0.27501041165787316827732023863, 0.846012620433440372408705899671, 1.03215239411056794736302919822, 1.20506266560044681773564449660, 1.23168300996773421820606583965, 1.62247224845569329710186986049, 1.68997328924518543236132507240, 1.70118809959118278879721452230, 2.03879200196406574979034740172, 2.22366744686821132209482183953, 2.38587209817188255723455286720, 2.41718873513721421987817455832, 2.80367136786194763605564137039, 2.82244357483652658911517576183, 2.90404489858447869689746762304, 2.90418213591286321479388772324, 3.11616100188518883379337162345, 3.22356086965437554225874724068, 3.59047065985743754673622987913, 3.65367636488729184176222526808, 3.66552266549810793932031855570, 3.74299674049081273310336159002, 3.87864838548261594036174125617, 3.94279230562842542456720748285
Plot not available for L-functions of degree greater than 10.
