| L(s) = 1 | + 4·7-s + 24·17-s − 12·23-s − 24·41-s + 12·47-s + 8·49-s − 32·73-s − 96·79-s − 8·81-s − 32·97-s − 28·103-s + 96·119-s − 48·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 48·161-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 5.82·17-s − 2.50·23-s − 3.74·41-s + 1.75·47-s + 8/7·49-s − 3.74·73-s − 10.8·79-s − 8/9·81-s − 3.24·97-s − 2.75·103-s + 8.80·119-s − 4.36·121-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 − 3.78·161-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.933754034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.933754034\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 34 T^{4} + p^{4} T^{8} \) |
| good | 3 | \( 1 + 8 T^{4} + 94 T^{8} + 8 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( ( 1 - 2 T + 2 T^{2} + 6 T^{3} - 82 T^{4} + 6 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 24 T^{2} + 302 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 62 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 8 T + p T^{2} )^{4}( 1 + 2 T + p T^{2} )^{4} \) |
| 19 | \( ( 1 - 8 T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4} \) |
| 23 | \( ( 1 + 6 T + 18 T^{2} + 102 T^{3} + 542 T^{4} + 102 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 64 T^{2} + 2190 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 2338 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 + 2632 T^{4} + 3107358 T^{8} + 2632 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 6 T + 18 T^{2} - 246 T^{3} + 3326 T^{4} - 246 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 2846 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 112 T^{2} + 9822 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 56 T^{4} - 39549474 T^{8} - 56 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 80 T^{2} + 4878 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 16 T + 128 T^{2} + 1008 T^{3} + 7838 T^{4} + 1008 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 12 T + p T^{2} )^{8} \) |
| 83 | \( 1 + 8200 T^{4} + 98353758 T^{8} + 8200 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 116 T^{2} + 7110 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 16 T + 128 T^{2} + 1392 T^{3} + 15038 T^{4} + 1392 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−5.35265042544788483900730385861, −5.29585085309259754788388190565, −4.96393915148140038987392594617, −4.79633566736593300562678079410, −4.47320010592608536206187818568, −4.42921306707228677405983253199, −4.27493227723504853500080653611, −4.07741614170489207705698863391, −4.00500071984377622151182081689, −3.92907818449291946471738704722, −3.86292682817289540769637025535, −3.37701364160062147997722977312, −3.19978138227757168644917766502, −3.09087052806274083177897050892, −3.01528621997394746819345891710, −2.81956669829911059749810591717, −2.70333672984345784861745328648, −2.61843304996480414060947674664, −1.97046487876769149429060984042, −1.69518821725856738390319557459, −1.51776662830399080839565989131, −1.48873337569911854326346201511, −1.40444699084317220501199822112, −1.17750516761386428149930797872, −0.29769479477600662475967474098,
0.29769479477600662475967474098, 1.17750516761386428149930797872, 1.40444699084317220501199822112, 1.48873337569911854326346201511, 1.51776662830399080839565989131, 1.69518821725856738390319557459, 1.97046487876769149429060984042, 2.61843304996480414060947674664, 2.70333672984345784861745328648, 2.81956669829911059749810591717, 3.01528621997394746819345891710, 3.09087052806274083177897050892, 3.19978138227757168644917766502, 3.37701364160062147997722977312, 3.86292682817289540769637025535, 3.92907818449291946471738704722, 4.00500071984377622151182081689, 4.07741614170489207705698863391, 4.27493227723504853500080653611, 4.42921306707228677405983253199, 4.47320010592608536206187818568, 4.79633566736593300562678079410, 4.96393915148140038987392594617, 5.29585085309259754788388190565, 5.35265042544788483900730385861
Plot not available for L-functions of degree greater than 10.
