L(s) = 1 | − 8·13-s + 16·25-s − 8·37-s − 56·73-s + 40·97-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-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 | − 2.21·13-s + 16/5·25-s − 1.31·37-s − 6.55·73-s + 4.06·97-s − 0.727·121-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 + 2.46·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.070403473\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.070403473\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
good | 7 | \( ( 1 + 2 T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 4 T + p T^{2} )^{4}( 1 + 6 T + p T^{2} )^{4} \) |
| 17 | \( ( 1 - 254 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 3598 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 2302 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2}( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + p T^{2} )^{8} \) |
| 67 | \( ( 1 - 7534 T^{4} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - p T^{2} )^{8} \) |
| 73 | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 146 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 80 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 18 T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.58914976665583933554462901865, −3.55226660263694279875391489962, −3.46185112242296767910533152157, −3.12211322290872932866149619177, −3.08500776460302401355617631349, −2.96894853134270743437548813413, −2.89211040006300963868471797534, −2.88056195324428767634892765513, −2.87548642022768993928350941229, −2.64033539227939357913588643296, −2.40627861305673002614371633324, −2.24464185671693851951835457702, −2.17821777130952702205467232807, −2.06689431670540556264248813808, −1.85051261872529498005956676953, −1.79236955736606320461160543112, −1.59683943184660485812465870582, −1.45007702003448595143497928063, −1.40589291838087897196198671755, −1.12351827514121929100446664217, −0.855969000286244316813069816720, −0.805249522424721214980345985212, −0.49660813925101686028135179649, −0.38479547858160083146220780828, −0.22726961510567736278823297481,
0.22726961510567736278823297481, 0.38479547858160083146220780828, 0.49660813925101686028135179649, 0.805249522424721214980345985212, 0.855969000286244316813069816720, 1.12351827514121929100446664217, 1.40589291838087897196198671755, 1.45007702003448595143497928063, 1.59683943184660485812465870582, 1.79236955736606320461160543112, 1.85051261872529498005956676953, 2.06689431670540556264248813808, 2.17821777130952702205467232807, 2.24464185671693851951835457702, 2.40627861305673002614371633324, 2.64033539227939357913588643296, 2.87548642022768993928350941229, 2.88056195324428767634892765513, 2.89211040006300963868471797534, 2.96894853134270743437548813413, 3.08500776460302401355617631349, 3.12211322290872932866149619177, 3.46185112242296767910533152157, 3.55226660263694279875391489962, 3.58914976665583933554462901865
Plot not available for L-functions of degree greater than 10.