| L(s) = 1 | − 8·4-s + 12·5-s + 48·16-s − 24·19-s − 96·20-s + 72·25-s − 98·49-s + 216·59-s − 24·61-s − 256·64-s + 192·76-s + 576·80-s + 62·81-s − 288·95-s − 576·100-s + 456·101-s + 484·121-s + 300·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2·4-s + 12/5·5-s + 3·16-s − 1.26·19-s − 4.79·20-s + 2.87·25-s − 2·49-s + 3.66·59-s − 0.393·61-s − 4·64-s + 2.52·76-s + 36/5·80-s + 0.765·81-s − 3.03·95-s − 5.75·100-s + 4.51·101-s + 4·121-s + 12/5·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.059841368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.059841368\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )( 1 + 4 T + 8 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 36 T + 648 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )( 1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2}( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 108 T + 5832 T^{2} - 108 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 2018 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 4418 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 36 T + 648 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )( 1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
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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
−8.485800754009883607946744035657, −8.388380105245402626473496476356, −8.104946585605901743523054974223, −7.59278272379683870839738427636, −7.42750663496746075811170641854, −7.09187283000678463428008831397, −6.75189554850861888903195968609, −6.33411948958469444269447632467, −6.14746410677005567460043785525, −6.05536527119852521981148437088, −5.64897067882349695251305370357, −5.50521714600397236987063425049, −5.01815115087204119561256123227, −4.88243964759236697288290498892, −4.79045247384708943620255980385, −4.16442733809072998996234787985, −4.08383353022252602641706966065, −3.57392114269976444739407280183, −3.15450669190579261230909154855, −2.97843694399853507727034249702, −2.14710593620478366639738313363, −2.02384707449424628895067689616, −1.70218016272876016319765340176, −0.858832909094274798254945025081, −0.55705475856640411378650876433,
0.55705475856640411378650876433, 0.858832909094274798254945025081, 1.70218016272876016319765340176, 2.02384707449424628895067689616, 2.14710593620478366639738313363, 2.97843694399853507727034249702, 3.15450669190579261230909154855, 3.57392114269976444739407280183, 4.08383353022252602641706966065, 4.16442733809072998996234787985, 4.79045247384708943620255980385, 4.88243964759236697288290498892, 5.01815115087204119561256123227, 5.50521714600397236987063425049, 5.64897067882349695251305370357, 6.05536527119852521981148437088, 6.14746410677005567460043785525, 6.33411948958469444269447632467, 6.75189554850861888903195968609, 7.09187283000678463428008831397, 7.42750663496746075811170641854, 7.59278272379683870839738427636, 8.104946585605901743523054974223, 8.388380105245402626473496476356, 8.485800754009883607946744035657
