| L(s) = 1 | − 4-s + 6·9-s − 3·16-s − 6·36-s + 28·49-s − 8·61-s + 7·64-s + 27·81-s − 56·109-s − 44·121-s + 127-s + 131-s + 137-s + 139-s − 18·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 28·196-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2·9-s − 3/4·16-s − 36-s + 4·49-s − 1.02·61-s + 7/8·64-s + 3·81-s − 5.36·109-s − 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3/2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 2·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.860502809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.860502809\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
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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
−8.834857352242505299862297906693, −8.072956056963245475517567963300, −8.042117752955831477681839279600, −7.83743197920996102622876190385, −7.53199024700368932420842637473, −7.14311312029625923470584536406, −7.09165322969349987606342131903, −6.79613763137732242710813556925, −6.56238621750688459368607035467, −6.12363996841608109918257343483, −6.11648504470573418558988894707, −5.37952273885941959794617987601, −5.24910132319504212874055274480, −5.21252859413873842000939297548, −4.70492407682606185113424540184, −4.22710936322728793186216207473, −4.09953679252022700648221542972, −4.00566856173883031188541835477, −3.74233123488470895105617752460, −3.03187788127953670040836800673, −2.58145881607263761644060066635, −2.45657223494028021055502165666, −1.69383047641562142343578541543, −1.43369041309139557867186498191, −0.68921781166120861241284768793,
0.68921781166120861241284768793, 1.43369041309139557867186498191, 1.69383047641562142343578541543, 2.45657223494028021055502165666, 2.58145881607263761644060066635, 3.03187788127953670040836800673, 3.74233123488470895105617752460, 4.00566856173883031188541835477, 4.09953679252022700648221542972, 4.22710936322728793186216207473, 4.70492407682606185113424540184, 5.21252859413873842000939297548, 5.24910132319504212874055274480, 5.37952273885941959794617987601, 6.11648504470573418558988894707, 6.12363996841608109918257343483, 6.56238621750688459368607035467, 6.79613763137732242710813556925, 7.09165322969349987606342131903, 7.14311312029625923470584536406, 7.53199024700368932420842637473, 7.83743197920996102622876190385, 8.042117752955831477681839279600, 8.072956056963245475517567963300, 8.834857352242505299862297906693
