| L(s) = 1 | + 4-s − 14·11-s − 4·16-s − 20·29-s + 4·31-s + 14·41-s − 14·44-s + 9·49-s − 22·59-s + 6·61-s − 5·64-s + 32·71-s − 4·79-s + 48·89-s − 6·101-s − 26·109-s − 20·116-s + 85·121-s + 4·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 4.22·11-s − 16-s − 3.71·29-s + 0.718·31-s + 2.18·41-s − 2.11·44-s + 9/7·49-s − 2.86·59-s + 0.768·61-s − 5/8·64-s + 3.79·71-s − 0.450·79-s + 5.08·89-s − 0.597·101-s − 2.49·109-s − 1.85·116-s + 7.72·121-s + 0.359·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.025097891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.025097891\) |
| \(L(\frac{3}{2})\) |
|
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 | $D_4\times C_2$ | \( 1 - T^{2} + 5 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 9 T^{2} + 37 T^{4} - 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 7 T + 31 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 41 T^{2} + 729 T^{4} - 41 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 61 T^{2} + 1505 T^{4} - 61 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 29 T^{2} + 1005 T^{4} - 29 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 2 T + 11 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 65 T^{2} + 2361 T^{4} - 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 7 T + 91 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 53 T^{2} + 3669 T^{4} - 53 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 82 T^{2} + 5891 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 25 T^{2} + 1793 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 11 T + 145 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 3 T + 121 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 234 T^{2} + 22459 T^{4} - 234 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 16 T + 193 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 194 T^{2} + 18195 T^{4} - 194 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 2 T + 146 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 190 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.43122171572815243172044200809, −6.36905518116049439960293597428, −6.20218930272920820786885564510, −5.73831721320107953240381115629, −5.56705892234801780091448150820, −5.42058170492753703990040132583, −5.20239204024948086474990512082, −5.12509252786266704348988130238, −4.91469231894279124640065687653, −4.79792490223273012707435783205, −4.24584732698004990495017217789, −4.06154291239750891742648637693, −3.94316841742954383791980346322, −3.70392715835056095711732195822, −3.34421537994002317136287657629, −2.96271982256561094231781750768, −2.77808129847246305525255870913, −2.64846090073802815257012922264, −2.41957268030263424394932881204, −2.13910819870554293294759976535, −1.85799974831376001280034916562, −1.81882972709373966295922788506, −1.02018502319151761949563514793, −0.52891907312726246828577165034, −0.26062051214093071115293430640,
0.26062051214093071115293430640, 0.52891907312726246828577165034, 1.02018502319151761949563514793, 1.81882972709373966295922788506, 1.85799974831376001280034916562, 2.13910819870554293294759976535, 2.41957268030263424394932881204, 2.64846090073802815257012922264, 2.77808129847246305525255870913, 2.96271982256561094231781750768, 3.34421537994002317136287657629, 3.70392715835056095711732195822, 3.94316841742954383791980346322, 4.06154291239750891742648637693, 4.24584732698004990495017217789, 4.79792490223273012707435783205, 4.91469231894279124640065687653, 5.12509252786266704348988130238, 5.20239204024948086474990512082, 5.42058170492753703990040132583, 5.56705892234801780091448150820, 5.73831721320107953240381115629, 6.20218930272920820786885564510, 6.36905518116049439960293597428, 6.43122171572815243172044200809
