| L(s) = 1 | + 8·17-s + 32·47-s − 4·49-s + 32·79-s + 14·81-s − 8·97-s − 8·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | + 1.94·17-s + 4.66·47-s − 4/7·49-s + 3.60·79-s + 14/9·81-s − 0.812·97-s − 0.752·113-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 + 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 + 0.0663·227-s + 0.0660·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.147852940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.147852940\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )( 1 + 8 T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 82 T^{4} + p^{4} T^{8} \) |
| 13 | $C_2^3$ | \( 1 + 146 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \) |
| 23 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^3$ | \( 1 - 1198 T^{4} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 - 2062 T^{4} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 1198 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 53 | $C_2^3$ | \( 1 - 718 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 878 T^{4} + p^{4} T^{8} \) |
| 61 | $C_2^3$ | \( 1 + 6482 T^{4} + p^{4} T^{8} \) |
| 67 | $C_2^3$ | \( 1 - 7822 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 + 3122 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 174 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
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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
−7.22701457878079858275664800472, −7.17292089608055547753521233140, −6.51572067217407206485432866094, −6.48855825524144250605597903977, −6.21706140665644193498539726895, −5.88403925839702206332039992215, −5.75580977862261704534476117709, −5.71825374667437010823562266658, −5.13313904663649993546507091065, −5.07846918071791264040512175711, −5.04301086149437414324540959060, −4.60051403689155093043264581036, −4.13990520583006906587128290044, −4.04885733417651694282974305745, −3.90012383579256147622707480175, −3.52195504667435645060392012954, −3.22418325584603891664000756420, −3.05820101254712006132722471359, −2.72603965828437921003271189657, −2.29047467704905323043920468610, −2.08258155784625936442467934634, −1.84150029290889241218656329183, −1.10078576728659743524075207070, −0.949978495414823838712843208652, −0.57041881947983786115844130950,
0.57041881947983786115844130950, 0.949978495414823838712843208652, 1.10078576728659743524075207070, 1.84150029290889241218656329183, 2.08258155784625936442467934634, 2.29047467704905323043920468610, 2.72603965828437921003271189657, 3.05820101254712006132722471359, 3.22418325584603891664000756420, 3.52195504667435645060392012954, 3.90012383579256147622707480175, 4.04885733417651694282974305745, 4.13990520583006906587128290044, 4.60051403689155093043264581036, 5.04301086149437414324540959060, 5.07846918071791264040512175711, 5.13313904663649993546507091065, 5.71825374667437010823562266658, 5.75580977862261704534476117709, 5.88403925839702206332039992215, 6.21706140665644193498539726895, 6.48855825524144250605597903977, 6.51572067217407206485432866094, 7.17292089608055547753521233140, 7.22701457878079858275664800472
