| L(s) = 1 | − 14·3-s + 81·9-s + 46·11-s + 1.14e3·17-s + 868·19-s − 1.25e3·25-s − 658·27-s − 644·33-s − 1.24e3·41-s + 3.50e3·43-s − 4.80e3·49-s − 1.60e4·51-s − 1.21e4·57-s + 238·59-s + 5.13e3·67-s − 1.90e4·73-s + 1.75e4·75-s + 9.21e3·81-s + 2.23e4·83-s + 2.18e4·89-s − 9.98e3·97-s + 3.72e3·99-s + 1.75e4·107-s + 3.18e4·113-s − 1.25e4·121-s + 1.74e4·123-s + ⋯ |
| L(s) = 1 | − 1.55·3-s + 9-s + 0.380·11-s + 3.97·17-s + 2.40·19-s − 2·25-s − 0.902·27-s − 0.591·33-s − 0.741·41-s + 1.89·43-s − 2·49-s − 6.17·51-s − 3.74·57-s + 0.0683·59-s + 1.14·67-s − 3.56·73-s + 28/9·75-s + 1.40·81-s + 3.24·83-s + 2.76·89-s − 1.06·97-s + 0.380·99-s + 1.53·107-s + 2.49·113-s − 0.855·121-s + 1.15·123-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4233905748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4233905748\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + 14 T + 115 T^{2} + 14 p^{4} T^{3} + p^{8} T^{4} \) |
| good | 5 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2}( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2}( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 - 46 T + p^{4} T^{2} )^{2}( 1 + 46 T - 12525 T^{2} + 46 p^{4} T^{3} + p^{8} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2}( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 574 T + 245955 T^{2} - 574 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 434 T + 58035 T^{2} - 434 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2}( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2}( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2}( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + 1246 T + p^{4} T^{2} )^{2}( 1 - 1246 T - 1273245 T^{2} - 1246 p^{4} T^{3} + p^{8} T^{4} ) \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 - 3502 T + p^{4} T^{2} )^{2}( 1 + 3502 T + 8845203 T^{2} + 3502 p^{4} T^{3} + p^{8} T^{4} ) \) |
| 47 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2}( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 - 238 T + p^{4} T^{2} )^{2}( 1 + 238 T - 12060717 T^{2} + 238 p^{4} T^{3} + p^{8} T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2}( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - 5134 T + p^{4} T^{2} )^{2}( 1 + 5134 T + 6206835 T^{2} + 5134 p^{4} T^{3} + p^{8} T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 9506 T + 61965795 T^{2} + 9506 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2}( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 11186 T + 77668275 T^{2} - 11186 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 5474 T + p^{4} T^{2} )^{4} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + 9982 T + p^{4} T^{2} )^{2}( 1 - 9982 T + 11111043 T^{2} - 9982 p^{4} T^{3} + p^{8} T^{4} ) \) |
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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
−7.74889500732210858074722733450, −7.64344180291878394200836884958, −7.53990900188019383666568971835, −7.26968957591172043645683944966, −6.90211967781255127227360640662, −6.29375678319169759746268148764, −6.29297317345642616684080321256, −5.99892849981497965787424791770, −5.67813527779654289210199092011, −5.65390579849961299942591104169, −5.26664166576470891249718143194, −5.06410449433913623244668312840, −4.86867189548431414692575278310, −4.47853005237686893696862724659, −3.77561777854955062618248444765, −3.66290009708246247923026939816, −3.57482118310107428508001765954, −3.09403727310752444187100358014, −2.90142778197081530168934670622, −2.13080570563785158937753288531, −1.84431876813087873274230298386, −1.18523074674987428107189915593, −0.979084626187478912535862947091, −0.968710453945814009096261376504, −0.11351613859563176430207225461,
0.11351613859563176430207225461, 0.968710453945814009096261376504, 0.979084626187478912535862947091, 1.18523074674987428107189915593, 1.84431876813087873274230298386, 2.13080570563785158937753288531, 2.90142778197081530168934670622, 3.09403727310752444187100358014, 3.57482118310107428508001765954, 3.66290009708246247923026939816, 3.77561777854955062618248444765, 4.47853005237686893696862724659, 4.86867189548431414692575278310, 5.06410449433913623244668312840, 5.26664166576470891249718143194, 5.65390579849961299942591104169, 5.67813527779654289210199092011, 5.99892849981497965787424791770, 6.29297317345642616684080321256, 6.29375678319169759746268148764, 6.90211967781255127227360640662, 7.26968957591172043645683944966, 7.53990900188019383666568971835, 7.64344180291878394200836884958, 7.74889500732210858074722733450
