| L(s) = 1 | + 4·19-s − 24·29-s − 4·31-s + 24·41-s + 20·49-s − 24·59-s − 16·61-s + 24·71-s + 16·79-s + 24·101-s + 4·109-s − 38·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 0.917·19-s − 4.45·29-s − 0.718·31-s + 3.74·41-s + 20/7·49-s − 3.12·59-s − 2.04·61-s + 2.84·71-s + 1.80·79-s + 2.38·101-s + 0.383·109-s − 3.45·121-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 + 1.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.457607946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.457607946\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 186 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 246 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 954 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4746 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6762 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 11034 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 5130 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 12 T + 151 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 67 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 2166 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 12 T + 151 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 188 T^{2} + 16794 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 8586 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 151 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 380 T^{2} + 54906 T^{4} - 380 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.57102585267548162113678575558, −5.43373247788282536580030267928, −5.14229331087035426504742197099, −4.83742244922653844135233795420, −4.76016505369932647820947080055, −4.72223964476221415109475682988, −4.38056870104588391142390359373, −4.00897604053955738836722710170, −3.85160082984493484948385184900, −3.83965080749752124721595933566, −3.75579654970022471664609909864, −3.41975297551013825950267292812, −3.34179194150733243741301246630, −2.99951132303552983284904893186, −2.71610249868806172257141623291, −2.43565002958516511346958203896, −2.28987874419423450495269966416, −2.25194070438566794164048200024, −2.02333429146801692597178282020, −1.59060936046664083177389947721, −1.29255634186707623467288052427, −1.09441886584919454489348562437, −1.08561613217301888949976881122, −0.39250421923980820701298727275, −0.19290917447958255108560666359,
0.19290917447958255108560666359, 0.39250421923980820701298727275, 1.08561613217301888949976881122, 1.09441886584919454489348562437, 1.29255634186707623467288052427, 1.59060936046664083177389947721, 2.02333429146801692597178282020, 2.25194070438566794164048200024, 2.28987874419423450495269966416, 2.43565002958516511346958203896, 2.71610249868806172257141623291, 2.99951132303552983284904893186, 3.34179194150733243741301246630, 3.41975297551013825950267292812, 3.75579654970022471664609909864, 3.83965080749752124721595933566, 3.85160082984493484948385184900, 4.00897604053955738836722710170, 4.38056870104588391142390359373, 4.72223964476221415109475682988, 4.76016505369932647820947080055, 4.83742244922653844135233795420, 5.14229331087035426504742197099, 5.43373247788282536580030267928, 5.57102585267548162113678575558
