| L(s) = 1 | + 4·9-s − 24·41-s + 28·49-s − 6·81-s + 72·89-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-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 | + 4/3·9-s − 3.74·41-s + 4·49-s − 2/3·81-s + 7.63·89-s + 2.54·121-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 − 4·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.220715367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.220715367\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 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
−6.04164770654386296906493366396, −5.84524222847774122420949423574, −5.84254365795268599510982252585, −5.54528868499144386230330889876, −5.29675726799086214205937317228, −5.15777430931850633397067813338, −4.71726027596348688226465133794, −4.68878493757624640262400859800, −4.56280077651647637084865446581, −4.53705384143802619088941301196, −3.95709800064914553825214580668, −3.76663012259756505261253472165, −3.57668309019101037114579375121, −3.50656957040484719941304829333, −3.41088168422958046804093214163, −2.89395330691229041411015576339, −2.71987037431857806562736523410, −2.21594337941838617067209921127, −2.14719491180285972853108765076, −2.08700221489135278025739649113, −1.66001857573302803271639112491, −1.30260998178410283389165241399, −1.01289868854126216052782011381, −0.77373207758132361208690340736, −0.27587985931878483352762435520,
0.27587985931878483352762435520, 0.77373207758132361208690340736, 1.01289868854126216052782011381, 1.30260998178410283389165241399, 1.66001857573302803271639112491, 2.08700221489135278025739649113, 2.14719491180285972853108765076, 2.21594337941838617067209921127, 2.71987037431857806562736523410, 2.89395330691229041411015576339, 3.41088168422958046804093214163, 3.50656957040484719941304829333, 3.57668309019101037114579375121, 3.76663012259756505261253472165, 3.95709800064914553825214580668, 4.53705384143802619088941301196, 4.56280077651647637084865446581, 4.68878493757624640262400859800, 4.71726027596348688226465133794, 5.15777430931850633397067813338, 5.29675726799086214205937317228, 5.54528868499144386230330889876, 5.84254365795268599510982252585, 5.84524222847774122420949423574, 6.04164770654386296906493366396
