| L(s) = 1 | − 18·3-s + 243·9-s − 284·11-s − 866·25-s − 2.91e3·27-s + 5.11e3·33-s + 4.79e3·49-s + 1.37e4·59-s + 1.63e4·73-s + 1.55e4·75-s + 3.28e4·81-s + 8.35e3·83-s + 3.45e4·97-s − 6.90e4·99-s + 3.10e4·107-s + 3.12e4·121-s + 127-s + 131-s + 137-s + 139-s − 8.63e4·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.71e4·169-s + ⋯ |
| L(s) = 1 | − 2·3-s + 3·9-s − 2.34·11-s − 1.38·25-s − 4·27-s + 4.69·33-s + 1.99·49-s + 3.94·59-s + 3.06·73-s + 2.77·75-s + 5·81-s + 1.21·83-s + 3.67·97-s − 7.04·99-s + 2.70·107-s + 2.13·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s − 3.99·147-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.010851931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.010851931\) |
| \(L(3)\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 866 T^{2} + p^{8} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4798 T^{2} + p^{8} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 142 T + p^{4} T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 745438 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 1618558 T^{2} + p^{8} T^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 5425438 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6862 T + p^{4} T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 8158 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 5237762 T^{2} + p^{8} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4178 T + p^{4} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 17282 T + p^{4} T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.89936676520173895518042239863, −10.45843670533051265785571825860, −10.21426302597950095241107870808, −9.954114447435316108219445216558, −9.298049213517266989150740255689, −8.590295934424410061520002728376, −7.82529204529493109841809035240, −7.68660826572365996501560769392, −7.14464630890318881440478803455, −6.57669001021009265254876950928, −6.02079843067190786234340706695, −5.56720684865002827822363715983, −5.12898112182448064256708397252, −4.94691566743387022448692144998, −4.03249557332999645313883846673, −3.62539034805980592831393327453, −2.33755625300288573238394760297, −2.07125367609498785507677329145, −0.71821875545580381753099668988, −0.51108800722134416947575983168,
0.51108800722134416947575983168, 0.71821875545580381753099668988, 2.07125367609498785507677329145, 2.33755625300288573238394760297, 3.62539034805980592831393327453, 4.03249557332999645313883846673, 4.94691566743387022448692144998, 5.12898112182448064256708397252, 5.56720684865002827822363715983, 6.02079843067190786234340706695, 6.57669001021009265254876950928, 7.14464630890318881440478803455, 7.68660826572365996501560769392, 7.82529204529493109841809035240, 8.590295934424410061520002728376, 9.298049213517266989150740255689, 9.954114447435316108219445216558, 10.21426302597950095241107870808, 10.45843670533051265785571825860, 10.89936676520173895518042239863
