| L(s) = 1 | + 2·4-s − 5·16-s − 8·19-s + 32·31-s + 20·49-s − 28·61-s − 20·64-s − 16·76-s − 8·79-s − 44·109-s − 20·121-s + 64·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 50·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 4-s − 5/4·16-s − 1.83·19-s + 5.74·31-s + 20/7·49-s − 3.58·61-s − 5/2·64-s − 1.83·76-s − 0.900·79-s − 4.21·109-s − 1.81·121-s + 5.74·124-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 + 3.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.595528167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.595528167\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 55 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 151 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 190 T^{2} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.51566896035934009893868840447, −6.36254490457928315677313940755, −6.28363569961192941036428684525, −5.95256969491012050268713111413, −5.69967325207386727261406353477, −5.62171107945741365620657903141, −5.09816717814197307883664807277, −5.07325631206401795385858994105, −4.55684446468755838420231779525, −4.53722064005510145192553450233, −4.38024644182308467261085656773, −4.29385597405379900920398936898, −3.99027584132772433523918590212, −3.74740179583538483522747141476, −3.26635679337664497482225881798, −2.85065386380197381335095136149, −2.75776967402327200214552043673, −2.69384006054574773533303821647, −2.60190979083153458403636109218, −2.01098669195593130483312995182, −1.96365714280001234364633736269, −1.48321088283183700866022534230, −1.17415526970609695804262739736, −0.800163558692170654301047797943, −0.28079742699792462762690322188,
0.28079742699792462762690322188, 0.800163558692170654301047797943, 1.17415526970609695804262739736, 1.48321088283183700866022534230, 1.96365714280001234364633736269, 2.01098669195593130483312995182, 2.60190979083153458403636109218, 2.69384006054574773533303821647, 2.75776967402327200214552043673, 2.85065386380197381335095136149, 3.26635679337664497482225881798, 3.74740179583538483522747141476, 3.99027584132772433523918590212, 4.29385597405379900920398936898, 4.38024644182308467261085656773, 4.53722064005510145192553450233, 4.55684446468755838420231779525, 5.07325631206401795385858994105, 5.09816717814197307883664807277, 5.62171107945741365620657903141, 5.69967325207386727261406353477, 5.95256969491012050268713111413, 6.28363569961192941036428684525, 6.36254490457928315677313940755, 6.51566896035934009893868840447
