| L(s) = 1 | + 4·3-s + 4-s − 2·9-s + 4·12-s − 4·16-s − 16·17-s + 12·23-s − 2·25-s − 40·27-s + 4·29-s − 2·36-s + 12·43-s − 16·48-s + 26·49-s − 64·51-s + 16·53-s − 4·61-s − 5·64-s − 16·68-s + 48·69-s − 8·75-s − 8·79-s − 55·81-s + 16·87-s + 12·92-s − 2·100-s + 36·101-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1/2·4-s − 2/3·9-s + 1.15·12-s − 16-s − 3.88·17-s + 2.50·23-s − 2/5·25-s − 7.69·27-s + 0.742·29-s − 1/3·36-s + 1.82·43-s − 2.30·48-s + 26/7·49-s − 8.96·51-s + 2.19·53-s − 0.512·61-s − 5/8·64-s − 1.94·68-s + 5.77·69-s − 0.923·75-s − 0.900·79-s − 6.11·81-s + 1.71·87-s + 1.25·92-s − 1/5·100-s + 3.58·101-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
| \(L(1)\) |
\(\approx\) |
\(3.314155367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.314155367\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - T^{2} + 5 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 10 T^{2} + 59 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 8 T + 37 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 42 T^{2} + 955 T^{4} - 42 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 29 | $D_{4}$ | \( ( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 61 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 73 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 6 T + p T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 5030 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 22 T^{2} - 6645 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 60 T^{2} - 1754 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 4 T + 110 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 220 T^{2} + 25046 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 250 T^{2} + 31259 T^{4} - 250 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 74 T^{2} + 12699 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.33783845192385672678610007151, −7.22906494741001589868184350980, −7.11750543823000065923678209129, −6.54615210084276495618198769249, −6.50498729328812230102940576659, −6.25447385130200532533504173511, −5.88406423917973855382087823922, −5.78237735914990167729217261077, −5.52035299801190722034314116864, −5.27961296603731997183166957700, −4.75462508972006341696082389192, −4.69568588763049536623614362532, −4.46615198507554606612259603737, −3.93929568582082845433700094046, −3.85044484808374319900907784407, −3.70913986405087788175847755240, −3.12614132770711678065679383745, −2.87673393773674537621593673624, −2.73929089197112560829296018160, −2.51561191865469427255836164011, −2.27669434407411454835486951992, −2.12010011695616190053804502294, −1.93123094981391850820771288660, −0.859888855228293190432234916455, −0.40549553316496237862697245364,
0.40549553316496237862697245364, 0.859888855228293190432234916455, 1.93123094981391850820771288660, 2.12010011695616190053804502294, 2.27669434407411454835486951992, 2.51561191865469427255836164011, 2.73929089197112560829296018160, 2.87673393773674537621593673624, 3.12614132770711678065679383745, 3.70913986405087788175847755240, 3.85044484808374319900907784407, 3.93929568582082845433700094046, 4.46615198507554606612259603737, 4.69568588763049536623614362532, 4.75462508972006341696082389192, 5.27961296603731997183166957700, 5.52035299801190722034314116864, 5.78237735914990167729217261077, 5.88406423917973855382087823922, 6.25447385130200532533504173511, 6.50498729328812230102940576659, 6.54615210084276495618198769249, 7.11750543823000065923678209129, 7.22906494741001589868184350980, 7.33783845192385672678610007151
