| L(s) = 1 | + 2·2-s + 3·4-s + 4·5-s − 4·7-s + 2·8-s + 4·9-s + 8·10-s − 4·11-s − 8·14-s + 4·17-s + 8·18-s − 4·19-s + 12·20-s − 8·22-s + 10·25-s − 12·28-s + 24·31-s − 6·32-s + 8·34-s − 16·35-s + 12·36-s − 8·38-s + 8·40-s + 12·41-s + 8·43-s − 12·44-s + 16·45-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.78·5-s − 1.51·7-s + 0.707·8-s + 4/3·9-s + 2.52·10-s − 1.20·11-s − 2.13·14-s + 0.970·17-s + 1.88·18-s − 0.917·19-s + 2.68·20-s − 1.70·22-s + 2·25-s − 2.26·28-s + 4.31·31-s − 1.06·32-s + 1.37·34-s − 2.70·35-s + 2·36-s − 1.29·38-s + 1.26·40-s + 1.87·41-s + 1.21·43-s − 1.80·44-s + 2.38·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.61375489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.61375489\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 13 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T + T^{2} + p T^{3} - 3 T^{4} + p^{2} T^{5} + p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 3 | $C_2^3$ | \( 1 - 4 T^{2} + 7 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 + 4 T + 6 T^{2} - 16 T^{3} - 61 T^{4} - 16 p T^{5} + 6 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T - 8 T^{2} + 8 T^{3} + 279 T^{4} + 8 p T^{5} - 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T - 14 T^{2} + 16 T^{3} + 339 T^{4} + 16 p T^{5} - 14 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 4 T - 24 T^{2} + 8 T^{3} + 935 T^{4} + 8 p T^{5} - 24 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 44 T^{2} + 1407 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 26 T^{2} - 165 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 - 2 T^{2} - 1365 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 34 T^{2} - 336 T^{3} + 4515 T^{4} - 336 p T^{5} + 34 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 8 T + 12 T^{2} + 272 T^{3} - 1897 T^{4} + 272 p T^{5} + 12 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 8 T^{2} + 216 T^{3} + 6519 T^{4} + 216 p T^{5} + 8 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 4 T - 32 T^{2} - 376 T^{3} - 3873 T^{4} - 376 p T^{5} - 32 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 4 T - 150 T^{2} + 112 T^{3} + 16595 T^{4} + 112 p T^{5} - 150 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.14787305973108074570996120835, −7.08066109336215834689609311092, −6.89359951301324352471513729260, −6.24052934961422595554113917239, −6.23473579132421504279672423367, −6.16560501548908950516357675065, −6.11410738487992635146134455495, −5.76510288896007166397074387518, −5.60184941456965970362800750404, −5.26096721842027611609768949040, −4.75414282311738684993206059577, −4.70957106681394500443948931938, −4.57522932446634176927990328914, −4.32846172225717505233816277063, −4.10011380258656297868106068090, −3.50558377018136617029428934228, −3.18040624542011556755735774814, −3.00162019023852644275567254205, −2.93754194993859792651876414987, −2.75463549871658897375841066997, −2.09408569909734086647969968976, −1.95889447291309011866925635794, −1.76557519005309632906145936520, −0.883307401964151858037114300272, −0.75417418157699704952913534633,
0.75417418157699704952913534633, 0.883307401964151858037114300272, 1.76557519005309632906145936520, 1.95889447291309011866925635794, 2.09408569909734086647969968976, 2.75463549871658897375841066997, 2.93754194993859792651876414987, 3.00162019023852644275567254205, 3.18040624542011556755735774814, 3.50558377018136617029428934228, 4.10011380258656297868106068090, 4.32846172225717505233816277063, 4.57522932446634176927990328914, 4.70957106681394500443948931938, 4.75414282311738684993206059577, 5.26096721842027611609768949040, 5.60184941456965970362800750404, 5.76510288896007166397074387518, 6.11410738487992635146134455495, 6.16560501548908950516357675065, 6.23473579132421504279672423367, 6.24052934961422595554113917239, 6.89359951301324352471513729260, 7.08066109336215834689609311092, 7.14787305973108074570996120835
