| L(s) = 1 | − 2·4-s − 2·9-s + 4·11-s + 3·16-s + 4·19-s + 28·29-s − 4·31-s + 4·36-s − 8·44-s + 8·49-s − 8·59-s − 4·61-s − 4·64-s − 8·71-s − 8·76-s − 4·79-s + 3·81-s + 4·89-s − 8·99-s + 8·101-s − 56·109-s − 56·116-s − 14·121-s + 8·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 4-s − 2/3·9-s + 1.20·11-s + 3/4·16-s + 0.917·19-s + 5.19·29-s − 0.718·31-s + 2/3·36-s − 1.20·44-s + 8/7·49-s − 1.04·59-s − 0.512·61-s − 1/2·64-s − 0.949·71-s − 0.917·76-s − 0.450·79-s + 1/3·81-s + 0.423·89-s − 0.804·99-s + 0.796·101-s − 5.36·109-s − 5.19·116-s − 1.27·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.396033547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.396033547\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 - 2 T + 13 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 24 T^{2} + 322 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 2 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 10 T^{2} + 923 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 14 T + 97 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 2 T + 23 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 3858 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 142 T^{2} + 9659 T^{4} - 142 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 2 T + 113 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 70 T^{2} + 6963 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T + 136 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 145 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 2 T + 119 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 134 T^{2} + 15027 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 2 T + 139 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 8 T^{2} - 17166 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.25490105919468537140233726145, −6.07624449038350617720235391477, −5.84892429667050632414652978631, −5.47088735701909635042179965730, −5.29542610633031579838208097504, −5.19953663323210909584148693012, −5.15343432393113033917436496034, −4.65395734749879077506681413749, −4.61067458523303326128441724806, −4.40572128027989097236587446500, −4.14084014610599374601723425945, −3.87040163386228343598810102587, −3.86070866446741393857175847510, −3.52198834978176609011506879076, −3.20594123463031410304060779007, −2.83460528379929129048581450809, −2.74337835302311460447436284519, −2.68796088148085169398372251055, −2.50854699029957035812777679753, −1.76961698298966174034915716717, −1.54002949925484883699636685543, −1.24870793459047439528013488688, −1.14311637964762292373356444725, −0.62169634352768602013473274433, −0.38470872338224236201417881905,
0.38470872338224236201417881905, 0.62169634352768602013473274433, 1.14311637964762292373356444725, 1.24870793459047439528013488688, 1.54002949925484883699636685543, 1.76961698298966174034915716717, 2.50854699029957035812777679753, 2.68796088148085169398372251055, 2.74337835302311460447436284519, 2.83460528379929129048581450809, 3.20594123463031410304060779007, 3.52198834978176609011506879076, 3.86070866446741393857175847510, 3.87040163386228343598810102587, 4.14084014610599374601723425945, 4.40572128027989097236587446500, 4.61067458523303326128441724806, 4.65395734749879077506681413749, 5.15343432393113033917436496034, 5.19953663323210909584148693012, 5.29542610633031579838208097504, 5.47088735701909635042179965730, 5.84892429667050632414652978631, 6.07624449038350617720235391477, 6.25490105919468537140233726145
