| L(s) = 1 | + 24·23-s − 4·25-s − 40·47-s + 24·49-s + 8·71-s + 40·73-s + 48·97-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 5.00·23-s − 4/5·25-s − 5.83·47-s + 24/7·49-s + 0.949·71-s + 4.68·73-s + 4.87·97-s + 4·121-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 + 0.0783·163-s + 0.0773·167-s + 3.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.604513217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.604513217\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 12 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 36 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 64 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
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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
−6.07354927287348495114561786600, −5.47009948471608903253990173633, −5.37386578189162678006975069305, −5.18869902695822134010980756291, −5.06357688760672510391974621345, −4.96057481412867214042343804655, −4.84353114784319437315436299077, −4.54633321999594124037276764559, −4.48236125230598524635269233080, −3.87671389386037801576592282539, −3.81739057146245058494164130206, −3.69697561829116945798302016457, −3.48622815491865014169050392003, −3.08386269687802162291920747145, −3.05214670725224450086655911930, −2.84663025818247163547277017920, −2.76794760390716487654127671663, −2.12405685590574694231427204884, −2.09108274125028759425641351592, −1.75495457466366195331429611955, −1.72807314767017120440826262292, −1.10001293584377675054322218066, −0.838369058726951553592484173434, −0.68225342563124183632341774407, −0.48992126553420974290398313524,
0.48992126553420974290398313524, 0.68225342563124183632341774407, 0.838369058726951553592484173434, 1.10001293584377675054322218066, 1.72807314767017120440826262292, 1.75495457466366195331429611955, 2.09108274125028759425641351592, 2.12405685590574694231427204884, 2.76794760390716487654127671663, 2.84663025818247163547277017920, 3.05214670725224450086655911930, 3.08386269687802162291920747145, 3.48622815491865014169050392003, 3.69697561829116945798302016457, 3.81739057146245058494164130206, 3.87671389386037801576592282539, 4.48236125230598524635269233080, 4.54633321999594124037276764559, 4.84353114784319437315436299077, 4.96057481412867214042343804655, 5.06357688760672510391974621345, 5.18869902695822134010980756291, 5.37386578189162678006975069305, 5.47009948471608903253990173633, 6.07354927287348495114561786600
