| L(s) = 1 | − 4·4-s − 8·9-s + 7·16-s − 4·19-s − 28·31-s + 32·36-s − 16·49-s − 12·59-s + 40·61-s − 8·64-s − 12·71-s + 16·76-s + 8·79-s + 33·81-s − 12·89-s − 36·101-s − 16·109-s + 112·124-s + 127-s + 131-s + 137-s + 139-s − 56·144-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 2·4-s − 8/3·9-s + 7/4·16-s − 0.917·19-s − 5.02·31-s + 16/3·36-s − 2.28·49-s − 1.56·59-s + 5.12·61-s − 64-s − 1.42·71-s + 1.83·76-s + 0.900·79-s + 11/3·81-s − 1.27·89-s − 3.58·101-s − 1.53·109-s + 10.0·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.66·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p^{2} T^{2} + 9 T^{4} + p^{4} T^{6} + p^{4} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 + 8 T^{2} + 31 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 16 T^{2} + 135 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 52 T^{2} + 1206 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 40 T^{2} + 1266 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 14 T + 108 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 100 T^{2} + 4806 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 79 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 16 T^{2} + 2439 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 40 T^{2} + 33 p T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 64 T^{2} + 1842 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 6 T + 124 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 20 T + 219 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 40 T^{2} - 2529 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 6 T + 124 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 6 T + 175 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 304 T^{2} + 40194 T^{4} + 304 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.52059389227283789311930290107, −6.47230313360010601980553456623, −5.93354786316324716849344633602, −5.83192743835226294669442025141, −5.70851098750357131062846009513, −5.42949726041299435500212274588, −5.41969858233680054287916381094, −5.25645998004365236980502174037, −5.12788491612687854836103645385, −4.71622270586471447740983193029, −4.46463399037072436410576721694, −4.46338634088067092376584306036, −4.03280359942784010004037806808, −3.86742807739526323050827512030, −3.69063286875781572344806895076, −3.39153282375293826501620624426, −3.36274344519274170373440833788, −3.16092801031452559673392835022, −2.70613256642045019692628759148, −2.32973856269040244328993633063, −2.30322949296656629952665844367, −2.11174103918127809538545989604, −1.62446689256832513050787406694, −1.13390773918921887309377951087, −1.12615663508548652106420889042, 0, 0, 0, 0,
1.12615663508548652106420889042, 1.13390773918921887309377951087, 1.62446689256832513050787406694, 2.11174103918127809538545989604, 2.30322949296656629952665844367, 2.32973856269040244328993633063, 2.70613256642045019692628759148, 3.16092801031452559673392835022, 3.36274344519274170373440833788, 3.39153282375293826501620624426, 3.69063286875781572344806895076, 3.86742807739526323050827512030, 4.03280359942784010004037806808, 4.46338634088067092376584306036, 4.46463399037072436410576721694, 4.71622270586471447740983193029, 5.12788491612687854836103645385, 5.25645998004365236980502174037, 5.41969858233680054287916381094, 5.42949726041299435500212274588, 5.70851098750357131062846009513, 5.83192743835226294669442025141, 5.93354786316324716849344633602, 6.47230313360010601980553456623, 6.52059389227283789311930290107
