| L(s) = 1 | + 2·3-s − 2·5-s − 2·7-s + 3·9-s − 8·13-s − 4·15-s − 2·17-s − 2·19-s − 4·21-s − 8·23-s − 4·25-s + 4·27-s − 2·29-s − 4·31-s + 4·35-s − 8·37-s − 16·39-s − 12·41-s − 6·45-s + 2·47-s + 3·49-s − 4·51-s + 14·53-s − 4·57-s − 8·59-s − 8·61-s − 6·63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 0.755·7-s + 9-s − 2.21·13-s − 1.03·15-s − 0.485·17-s − 0.458·19-s − 0.872·21-s − 1.66·23-s − 4/5·25-s + 0.769·27-s − 0.371·29-s − 0.718·31-s + 0.676·35-s − 1.31·37-s − 2.56·39-s − 1.87·41-s − 0.894·45-s + 0.291·47-s + 3/7·49-s − 0.560·51-s + 1.92·53-s − 0.529·57-s − 1.04·59-s − 1.02·61-s − 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2547216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2547216 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 92 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 14 T + 152 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 10 T + 140 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 172 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 202 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.049756608322081150738194287119, −8.937904194160606182488496591659, −8.269437357771425466382722793017, −8.151184128319022201781218678710, −7.50335456242339517333425498000, −7.40704885167260961065635383357, −6.85319567342577352402456201956, −6.75756065243963087479864872555, −5.77389054622750536507354432462, −5.68748622098255744291745446945, −4.83145021884166292200148545044, −4.55381018080794315093605473201, −3.87618389872982774635788888351, −3.81769987381258158511808038358, −3.12755645889295944342395056228, −2.65895010302699426090860535190, −2.06685566834651565159805977056, −1.74629627466307893717266668862, 0, 0,
1.74629627466307893717266668862, 2.06685566834651565159805977056, 2.65895010302699426090860535190, 3.12755645889295944342395056228, 3.81769987381258158511808038358, 3.87618389872982774635788888351, 4.55381018080794315093605473201, 4.83145021884166292200148545044, 5.68748622098255744291745446945, 5.77389054622750536507354432462, 6.75756065243963087479864872555, 6.85319567342577352402456201956, 7.40704885167260961065635383357, 7.50335456242339517333425498000, 8.151184128319022201781218678710, 8.269437357771425466382722793017, 8.937904194160606182488496591659, 9.049756608322081150738194287119
